VENICE, JUNE 11 -- Even Marco Polo, the 13th-century Venetian who made sightseeing an art form for generations ever after, would have to be impressed.

Here -- where Canaletto, Tintoretto, Titian and Veronese painted masterpieces still drawing crowds, where Monteverdi and Vivaldi composed music still being played and Goldoni wrote comedies of manners still being performed, where Casanova established philandering as a way of life still being practiced -- Nancy Reagan has managed what multitudes of pilgrims might have thought undoable. She has "done" Venice in one hour -- thanks to the efforts of uncounted staffers detailed by the White House, the State Department, the Defense Department, the Secret Service, the U.S. Information Service and no one has yet tallied exactly how many others.

"It's something of a logistical miracle," says Mrs. Reagan's press secretary, Elaine Crispen, who has been involved in the media planning since it began in earnest last winter.

Today Mrs. Reagan toured St. Mark's Cathedral and, at a kind of mountains-to-Mohammed exhibition arranged for her, viewed samples of Venetian crafts and culture at the Doges' Palace, then glided into camera range in a gondola for photographers positioned on the Bridge of Sighs. Once the pictures were snapped, she boarded a U.S. Navy launch that took her for a short ride along the Grand Canal to lunch at Harry's Bar.

The event and Mrs. Reagan's other appearances in Stockholm this week were the culmination of hundreds of transatlantic calls and thousands of man-hours spent in meetings and on on-site inspection trips. And in the end, they will cost taxpayers a hefty sum, as will the president's official activities. Those costs have not been added up here.

Crispen says she has "no idea" what the total package will cost.

"Clearly it will cost hundreds of thousands," says presidential spokesman Marlin Fitzwater. "But the costs are split up among so many budgets, it's impossible to come up with an exact figure."

The attention to detail is for Mrs. Reagan's benefit, but also for the media accompanying her. White House image-makers discovered early in Reagan's first term that Mrs. Reagan can be a valuable headline-getter, particularly if the news the president is making is less than favorable.

So her East Wing aides go along with advance teams from the president's West Wing and from the State Department to look out for the first lady's interests.

"The president's advance people may not be so sensitive to her needs," says a presidential aide, who also worries that the president's needs might sometimes be overlooked.

Today, for instance, as Mrs. Reagan walked along the carpeted aisle in the cathedral, her lead advance man softly warned, "There's a big dip here." She stepped over it without incident. Her advance team, however, was not at the Vatican Saturday, so there was no such warning about the carpet and she nearly fell.

"If a step to a platform is too high and someone forgets to warn Reagan, he might trip and be injured," the Reagan aide says. "One of our mistakes can mark the man forever, leaving him open to such speculation that he is 'too old' or 'ill.' "

It is a world in which appearances, of course, are everything.

It was no accident, for instance, that pictures taken of Mrs. Reagan and her party today show her with the cathedral in the background at St. Mark's Square, a square Napoleon called "the most beautiful drawing room in Europe," a place even Hitler said should be left alone.

"The advance team has to make sure there's something in the background that denotes where she is," says a former White House aide. "Otherwise it can be just any place."

Another concern is that "the principal" -- Nancy Reagan -- not be hidden by other guests or officials. So it was that today a White House aide put an X of masking tape on a cobblestone to stake out Mrs. Reagan's place to stop for photographers, but left no such marks for the mayor of Venice, the wife of Italy's acting prime minister and the wife of the U.S. ambassador, among others. Venice's semisacred pigeons apparently knew enough to stay away.

100,000 Miles in Six Years This is Mrs. Reagan's 12th foreign trip as first lady. Eight of those foreign visits were designated as part of her campaign against drug abuse; the others were ceremonial (two trips to Great Britain for royal weddings, one to Monaco for Princess Grace's funeral, one to Mexico to express U.S. support after the earthquake). Altogether, White House officials say, Mrs. Reagan has traveled more than 100,000 domestic and foreign miles in her campaign against drug abuse during the six years of the Reagan presidency.

The White House does not like surprises, which may explain in part why advance teams seem to saturate a site for days ahead of time.

Last fall, for example, Mrs. Reagan went to Harpers Ferry to welcome a trainload of comic-strip characters making a whistle-stop cross-country tour to warn against drug abuse. She made the trip from Washington in a White House limousine, and the visit itself lasted less than two hours.

A six-person advance staff had preceded her by four days. They stayed at the Bavarian Inn (whose rates to ordinary tourists are $90 to $105 a night) in Shepherdstown, W. Va., and for a while coowner Carol Asam thought Mrs. Reagan might also be a guest, since the White House had asked that additional rooms be blocked off.

Those plans changed, however, when it became apparent that Mrs. Reagan's motorcade could make the trip in 1 1/2 hours.

Reasons to Plan Ahead "The first lady has to have the support staff," says Crispen, "and it takes lodging them and flying them here if she's going to do something while she's here."

Crispen says she doubts very much if Mrs. Reagan is "totally aware of all the numbers that it takes" in dollars and people to get her where she's going.

"I don't think any president or first lady really knows all the details of what happens behind the scenes -- how many people are here and when they're all coming. She never asks, 'When is the West Wing going? When is the East Wing going?' The advance office schedules all that," Crispen says.

Crispen says it would be "negligent" on the part of the White House not to arrange something for Mrs. Reagan to see of the country she is visiting.

"In terms of diplomatic relations, I think countries are proud to share with visiting first ladies things they take special pride in. Mrs. Reagan is always interested in seeing them," Crispen says.

As it turns out, Mrs. Reagan was one of only two first ladies at this seven-nation economic summit, a get-together of leaders from industrialized nations that this year points up the transitory nature of power. Japan's Yasuhiro Nakasone is a lame duck, Great Britain's Margaret Thatcher had to return home for today's elections, France's Franc ois Mitterrand faces elections next year and Italy's Amintore Fanfani is only a caretaker prime minister until the election here Sunday.

Mila Mulroney, wife of Canadian Prime Minister Brian Mulroney, came to Venice also, but her path did not publicly cross Mrs. Reagan's. In contrast with Mrs. Reagan, whose full-time three-person Venice advance team was assisted by four people from the U.S. Embassy in Rome (plus support from other agencies when needed), Mrs. Mulroney had a grand total of one staffer advancing her trip.

Crispen says she knows of no way to measure the effect of Mrs. Reagan's visit, but that if the first lady had not made the effort, "I would think there would be hurt feelings."

The idea of Nancy Reagan staying home in Washington never crossed Crispen's mind, she says. "I don't even think of that," Crispen says. "I just think of them as a couple."

Certainly Ronald Reagan is happier when he and Mrs. Reagan are traveling "as a couple." Their staffs are well aware of that -- a case in point being the king-size bed flown last week from Portugal to Villa Condulmer, where the Reagans rested last week before the summit began. The Italian press especially loved the story of the imported bed, and the so-called "reports" of long showers, massages and hand-holding on strolls through Villa Condulmer's garden. The Italian media took speculation even further by calling the Reagans' stay in the country outside Venice "a belated honeymoon."

"Yes," Mrs. Reagan concurred when asked about it later, "it is like a honeymoon," though she denied the reported specifics.

Advancing in Waves Although from the outside it at times appears to be a junket in the grand old tradition of Washington junkets, those who advance a trip for the president or first lady say it's no honeymoon for them.

To prepare the way for Nancy Reagan on this trip, countless embassy and USIS personnel in Stockholm also lent assistance to the first lady's East Wing staff, as did embassy and USIS personnel working on the president's arrangements here in Venice.

For foreign trips like this, there are routinely three waves of advance teams: the site survey team, the preadvance team and the advance team, the largest group and the one that stays the longest. Mrs. Reagan's third team set up shop in Stockholm and Venice May 24, 10 days in advance of the Reagans' June 3 arrival in Europe.

Heading the Stockholm operation was Frank McKaye, an employe of Housing and Urban Development who is often detailed to the White House to assist on Mrs. Reagan's trips. Known as the "lead advance," McKaye coordinated final arrangements on sites to be visited, and worked out transportation, hotel and other logistical needs of Mrs. Reagan and her entourage. Working with him was Rick Holt, a Washington businessman working here as a volunteer. Heading the press advance was Betsy Koons, a deputy press secretary to Mrs. Reagan, assisted by a Social Security Administration assistant commissioner, Leslie Goodman, who, like McKaye, is on loan from her agency for the trip.

In Venice, Dennis Alfieri, a Los Angeles businessman and veteran at working Reagan trips, headed the advance team. Working with him as press advance has been Barbara Cook Fabiani of Great Falls, a former deputy press secretary to Mrs. Reagan. They, too, are volunteers, whose expenses are paid by the government but who receive no salary.

Trip trouble-shooter in charge of Mrs. Reagan's advance has been Marty Coyne, a quiet, unassuming man who is resigning at the end of this assignment to enter private business. To him fell the chore of signing off on all general logistics, including transportation and motorcades, hotel arrangements, and maintaining liaison with the Secret Service, embassy and foreign officials.

The first lady's chief of staff usually makes one preadvance trip in order to provide additional insight into decisions to be made on what to see and do and whom to meet. However, Jack Courtemanche, Mrs. Reagan's current chief, is less fascinated with the esthetic effects of a site or setting than was one of his predecessors. James Rosebush, who often read the latest Vogue magazine on the first lady's plane to keep up with one of her principal interests, felt obliged to advise her on colors to wear.

"He knew she wouldn't want to wear a red dress in a room with red flocked wallpaper," says one source.

The Image-Keepers Mrs. Reagan's appearance is of particular concern to two members of her traveling entourage, who stay where she and the president stay, whether it's Villa Condulmer or Windsor Castle: Anita Castelo, the first lady's personal assistant in charge of her some three dozen changes of outfits; and Julius Bengtsson of Los Angeles, who as First Hairdresser has seen her through the humidity of Indonesia and the icy winds of Lake Geneva.

In addition, Mrs. Reagan's party always includes her chief of staff, her press spokesman, the special assistant who serves as her personal secretary, a White House photographer, a medic, a State Department protocol officer and a detail of Secret Service agents whose round-the-clock replacements have preceded them to the destination. The size of the Secret Service detail is not released for security reasons.

In Stockholm, her personal entourage also included the head of the Secret Service, a National Security Council representative and the president's new adviser on drug abuse policy.

Mrs. Reagan's flight from Venice to Stockholm was made aboard Executive One, an Air Force Boeing 707 that is outfitted to accommodate the first lady and 40 to 44 other passengers as well as the pilots, stewards and military communications personnel. Traveling with her was a contingent of about a dozen reporters, photographers and television technicians whose news organizations will be billed later by the White House for the equivalent of coach fare plus $1 -- which for the Stockholm leg of Mrs. Reagan's trip was $1,208 round-trip. Some news organizations make separate travel arrangements when they're covering the Reagans, but reporters traveling on commercial planes chartered by the White House for the press -- as did about 200 from Washington to Venice -- pay a pro rata share of the exact flight costs. Hotel and ground transportation costs are handled similarly.

Dollars and Lire Unlike most tourists, who budget their travel dollars, President and Mrs. Reagan leave such details to others, of course. The expenses on foreign trips by the president and first lady are paid by various government agencies: the State Department, the USIS, the Secret Service, the Department of Defense, the White House Communications Agency.

The State Department's per diems differ for each city, depending on the costs of living there. For Venice -- acknowledged to be one of the more expensive, with an 18 percent tax on hotels to boot -- it is $175. Staff and volunteers working here were issued half of that in cash at the start of the trip, out of which they had to pay for their meals and incidentals.

The rest of the per diem is earmarked for hotel bills, which the State Department pays. And in Venice, that bill will be hefty. There are special government/corporate rates at the Cipriani, where the Reagans and their top personal and administrative aides are staying, but regular rates for its best rooms are $500 a night. At the Excelsior, where most of the press and some White House aides are staying, the regular rates are $300 to $350 a night.

The government also must pay for chartered ground and water transportation so staffers can get around, and for aides who must travel on commercial carriers at prevailing round-trip fares.

The Stockholm Segment In Stockholm, where aides said the schedule of 11 events -- appearances ranging from a 10-minute arrival ceremony, through a 25-minute drug discussion, to a three-hour official dinner -- in 2 1/2 days was the most intensive yet of Mrs. Reagan's single-city foreign trips, a small army awaited her. It comprised an estimated 100 Americans mobilized by the U.S. Embassy, USIS and the Secret Service at the request of the White House.

The operation went so smoothly that not even egg-throwing demonstrators protesting American foreign policy at various stops Mrs. Reagan made around the city affected her schedule or her spirits.

"I didn't see any of them," she said later. "It's become a way of life, hasn't it. If it happens it happens and you're not surprised."

Security precautions both frustrated and infuriated many Swedes. And while newspapers ran huge blowups of Mrs. Reagan dancing and singing with Swedish pop stars, they also showed her flanked by Secret Service agents, who were pointed out by circles drawn around their heads.

"We thought they were halos," was the inevitable joke making the rounds of Mrs. Reagan's entourage.

If Mrs. Reagan's stated mission was to examine Sweden's drug abuse programs, Swedes read an unstated mission as diplomatic. With the spring had come a thaw in once-frosty relations that went back to the Vietnam war.

Prime Minister Ingvar Carlsson's unscheduled drop-by at a photo opportunity set up for Mrs. Reagan with his wife Ingrid may have surprised the media but not the White House.

"That happens a lot," said Coyne. "He was probably sitting in his office and his wife invited him."

The Visit in Venice Until the Reagans' plans for a state visit to Rome before the summit fell through because of the forthcoming Italian elections, a schedule was being put together there for Mrs. Reagan. Her Rome advance team subsequently was disbanded, putting new emphasis on her activities here and in Stockholm.

In Venice, an ad hoc committee of lively Venetians calling themselves the Comitato Veneziano Per Gli Scambi Culturali Con Gli Stati Uniti d'America (roughly, the Committee of Cultural Exchanges Between the United States and Venice) had already been enlisted by U.S. Embassy officials to lend a hand. "For us, she's the first lady of the world," says Piero Mainardis de Campo, designer of furniture, buildings and cities, and one of the committee's movers and shakers. "If she feels happy here, it's a wonderful thing for Venice."

The event that finally emerged was a carefully condensed outing that began at 11:15 a.m., 15 minutes behind schedule, with Mrs. Reagan's arrival at St. Mark's Cathedral, where barricades had been erected to keep out other tourists.

For the next 15 minutes she toured the cathedral, where she had a wish ready for the Madonna, and then took a 10-minute tour of the arts and crafts exhibit at the Doges' Palace, and watched a 30-minute performance by Venetian singers and dancers in the palace square. That was followed by her first gondola ride, a 10-minute trip with a Secret Service agent and the gondolier, trailed by two other gondolas filled with singers and a clutch of rubber boats carrying police frogmen. The motor launch sent from the U.S. 6th Fleet in Naples whisked her to Harry's Bar in about five minutes. There she dined with Mrs. Fanfani; Ruth Rabb, wife of the U.S. ambassador here; and Marella Agnelli, of the Fiat Agnellis.

The Occasional Glitches It's trite but true: Even the best-laid plans go awry. There are invariably glitches.

For example, in 1984 in Shanghai, where Nancy Reagan visited a school before going on to a Buddhist monastery, vans driven by Chinese drivers left without the press despite repeated instructions that they were to turn around and come back to pick up reporters. On the same trip, the entire press corps found itself locked inside the pressroom by Chinese officials who were worried about security, and so was unable to cover the Reagans welcoming their Chinese hosts to a dinner in their honor.

On this week's trip the presidential bed shipped from Portugal to Villa Condulmer last week on orders from presidential assistant William Henckle took on superglitch status when the press found out and had a field day with it. Mrs. Reagan decreed that the bed not be moved to their hotel in Venice. Henckle, who oversees trip advance for the president, managed to survive Mrs. Reagan's displeasure, but the incident had its trickle-down effect on the first lady's own advance team, caught in the middle by an inquiring media.

Even the now-traditional "class photo" taken of Reagan with the members of the two Reagans' advance teams didn't work out quite right. It coincided precisely with Mrs. Reagan's one scheduled hour here today, the only time her team couldn't be present.