They came to Ronald Reagan's inauguration from across the nation yesterday, women wrapped in elegant furs, excited children clutching Muppet dolls, pot-bellied men smoking foul-smelling cigars -- a diverse slice of America.

But by dusk, nearly all 500,000 of them shared a common bond: a deep sense of patriotic pride -- the kind that makes eyes glisten and makes strangers gathered at the 40th president's swearing-in ceremony join hands and sing "America the Beautiful" from their hearts. For at least a day, everything seemed to be going right for America.

At last, the American hostages were free, the emasculating ordeal over. The noon-time sun splashed down on a spit-shined Washington filled with brass banks, blaring trumpets, glistening marble buildings and red, white and blue banners. And there was a new president promising a new beginning for America -- a reawakening, Reagan called it, to the days when Americans were as dynamic as their dreams.

"I'm so proud to be an American," exclaimed Wanita Douglas, a 58-year-old Denver resident, as she waved a tiny American flag at the corner of 14th Street and Pennsylvania Avenue. "I'm so proud to be an American."

The same words were repeated over and over again in a hundred different ways by a thousand different people as inaugural day continued with only one major snafu. The stubborn and overloaded Farecard system on the Metro subway simply couldn't cope with the thousands who wanted to ride the subway to see the swearing-in ceremony at the Capitol, the parade along Pennsylvania Avenue to the White House and the late-afternoon fireworks at the Capitol and Washington Monument.

For many of the spectators, even veteran inaugural officials, Reagan's celebration in record-setting 56-degree temperatures seemed almost magical, in part because of the release of the American hostages from Iran.

The crowd was reported to be much larger than the 350,000 who attended Jimmy Carter's "Y'all Come People's Celebration" in 1977, and the 300,000 on hand in 1969 when Richard M. Nixon was inaugurated, but far short of the 1.2 million who came to Lyndon B. Johnson's inaugural 16 years ago.

Americans are known for being impatient, but when yesterday's parade still hadn't started an hour after it was supposed to, a small group of spectators near President Reagan's reviewing stand in front of the White House began singing "God Bless America" rather than complaining. Soon, a cluster of them were singing patriotic songs.

Earlier, outside the Federal Trade Commission, a young, long-haired man wearing a red beret and Communist Workers Party pin had started to denounce Reagan and criticize the United States when a man wearing a three-piece business suit suddenly ripped an upsidedown American flag off the demonstrator's back.

"I just had to do it," said the man, who refused to give his name. "I just couldn't let him do that without doing something."

"There's something about being here," said Debra Simon, as she stood near the West Front of the Capitol, where Reagan took the oath of office at noon. "It's a chill that goes through you. All of the sudden you realize you're an American."

There were plenty of reminders. Flags draped the West Front of the Capitol, lined Pennsylvania Avenue and covered the huge tower clock at the old Post Office. Reagan's parade, with hundreds of marchers wearing costumes designed to tell this country's history, kept the crowds -- lined five to six deep along the street -- on their tiptoes. Many used red-and-white paper periscopes to see the more than 8,000 marchers and musicians, Alaskan sled dogs, equestrian teams and military units parade by.

After it was over, thousands of spectators listened to more band music outside the Capitol. And when the rousing music ended, the fireworks began, creating sequins of brilliant bursts above the sparkling Capitol.

"It's indescribable," said one watcher. "It's the best light show I've ever seen," added District resident Mike Coleman. "This is more than a political event," Coleman said, while watching yellow and green fireworks illuminate the Capitol dome. "It's an emotional feeling that overpowers the senses."

A finale of red bursts was accented by military bands which immediately struck up Aaron Compland's "Fanfare for the Common Man."

Shortly after 8:30 last night, Reagan and his wife Nancy plunged into their scheduled whistle stop tour of nine official inaugural balls with a visit at the Veterans Ball at the Capital Hilton Hotel. An hour later, he stopped by the ball at the Washington Hilton.

Throughout the nights, before retiring to the White House about 12:45 a.m., the Reagans kept up a hectic pace that kept them ahead of a schedule planned to the minute.

The festivities at these balls, climaxing the most expensive inaugural in American history, were telecast to 87 satellite balls in 37 states, where partygoers danced to the same music and saw the same sights ballgoers here saw, thanks to an elaborate $1.5-million closed circuit television system.

Complaints about lost tickets or oversold events that have plagued the inauguration cropped up at some of the balls last night, with some persons claiming they could not get into events for which they had paid.

And, while some people complained about the cost of the $8-million-plus Hollywood-style extravaganza, even the harshest critic admitted that "these Republicans really know how to throw an inauguration. It was impressive."

For most of yesterday's spectators, however, the day began less impressively than it ended.

The Metro Farecard system flubbed so badly that attendants at most subway gates simply allowed passangers to go through ticket gates free of charge by 10 a.m. Farecard machines broke down. Confused out-of-towners clogged subway stops. Slow ticket-taking machines at subway gates forced passengers to line up 10 abreast.

Once passengers got through the gates, they crowded dangerously close to the tracks, officials said. Even getting aboard was troublesome.

At one point, Metro officials told passengers at the Capitol South stop that "the trains are overloaded and the doors are sticking, so please try to give us a break." The mezzanine level was vibrating like and earthquake, passengers said.

The Reagans began their morning at a private church service, followed by a limousine ride with President Carter and his wife Rosalynn to the swearing-in. While they worshiped at St. John's Church, across Lafayett Square from the White House, crowds already were beginning to form on the parade route.

"I got laid off at Ford last week," said Paul Lewandowski, 42, of Dayton, Ohio, waiting on the avenue. "I spent the last money that I have for entertainment to come down here today. I just wanted to see what Reagan had to say."

By noon, more than 100,000 persons had gathered for the swearing in at the Capitol's West Front. It was the first time the swearing-in has been held on the West Front and judging from the reaction to it, the new location was popular.

"This [Reagan's speech] was fantastic," said Gene Miller, a goverfnment worker from San Antonio, Tex. "The man has the idea that we need to get government off our backs, that we need to defend ourselves. I think he can do it, and I'm going to help."

Susan and David Edwards of Winnetka, Ill., were even more impressed by Reagan. "We're going to call our baby, Ronnie," said Mrs. Edwards. "Because we want him to alwasy remember that he was named for a great American. I think the country is really about to turn around."

Vice President George Bush was sworn in first, and Reagan took his oath of office at 11:55 a.m. Reagan's swearing-in was followed by a traditional 21-gun salute that echoed around Capitol Hill and sent frightened pigeons flutting from the trees surrounding the Capitol's west lawn. Applause for Reagan rippled through the crowd, from the podium on the Capitol steps down the hill to the foot of Pennsylvania Avenue.

After the swearing-in just before noon, the new president attended a luncheon with top Congressional leaders in Statuary Hall of the Capitol, at which he announced the departure of the 52 hostages from Iran. About 1 1/2 hours after its scheduled 2:15 p.m. stepoff, the inaugural parade was ready to begin.

Pennsylvania Avenue, the city's main street and the avenue of the presidents, was ready and waiting long before the open-topped limousine carrying the new president slowly rolled onto the parade route from Capitol Hill.

As early as 9 a.m., the avenue had begun to put on its inaugural face. Bus loads of Capitol Hill, D.C. and Military police assumed their stations and scores of military men and women, shoes and brass shining, passed Henderson and gathered at checkpoints.

Secret Service men peered down from rooftops of government buildings. As waves of pedestrians headed down Pennsylvania Avenue to catch a glimpse of the swearing-in ceremony, police and military helicopters zoomed overhead and squadrons of policemen on motor scooters buzzed down the avenue, passing Vicki Ferguson seeking signatures for her pro-ERA positions.

"People are polite, but then a few are harsh and say nasty things," said Ferguson, an Alexandria resident. She paused and asked two women in full length mink coats for signatures but got a nasty reply."One of them said 'Absolute not,' and the other said 'F. . . ERA,' which surprised me, coming from women wearing mink coats."

Only the National-Organization for Women's pro-Equal Rights Amendment demonstration attracted much attention. Hundreds of white pennants with ERA YES printed in green on them could be seen along the parade route and volunteers like Eileen Sparks of Falls Church sought signatures on petitions in favor of the ERA.

Numerous demonstrations were scheduled, but fewer than 1,000 protesters showed up and they were viewed with more curiosity than for any political statement they wee making. A couple, walking around with tin tubes on their heads that had MX written on them, wandered along the parade route. A man dressed in a monk's costume carrying a sign which proclaimed "The End Is Near" walked beside another man wearing prison stripes and a Richard Nixon mask. Three young District residents, who said they are members of the People for the Ethical Treatment of Animals, held a banner that read "Take back your mink, take back your veal, what made you think, animals don't feel."

There were only three arrests, all for minor incidents.

Joggers dodged the walkers along the avenue and young soldiers, white-wall haircuts and acne, stood at parade rest checking out the giggling teen-aged girls while the giggling teen-aged girls checked out the soldiers.

All along Pennsylvania Avenue, the wealth of America passed: blacks, whites, Asians, Mexican-Americans, the rich intheir three-piece suits and furs, and the poor in bib overalls and torn jeans. They wore ski caps, cowboy hats, and no hats. Dignitaries in limousines passed them all.

"I'm glad t see the return of elegance to the White House," said Lonnie La Cour, 45, a widow from Rising Sun, Md., and a part-time disc jockey at a Quantico radio station, who made the 1 1/2-hour trip alone. Her red hair was mixed with gray, and she wore an American flag pin and a Ronald Reagan button. "We've had our fill with non-elegance. fPeople growing up today want something to look up to. Nobody dresses up anymore.

"I don't belong to that fancy group," she said, referring to the glitterati that rode by in chauffered linousines, "but we have to have some style in our culture."

Patiently, the hordes waited for the parade's start, and when it did, they greeted both Reagan and Bush with a sea of smiles, three and four deep along the curbs.

Between 9th and 10th Streets, where a group of demonstrators were concentrated in front of the FBI Building, Reagan and his wife, Nancy, sat down in the limousine and the car sped up. A few hundred feet along the route, the President and the First Lady stood up again.

Parents hoisted children onto their shoulders, and thousands waved American flags and bobbed ballons as floats and marching bands passed in review.

But there were many other signs and banners hoisted by those promoting other "liberal" causes, sending messages to this Republican administration despite its conservative orientations. Some signs read, "End the Arms Race," "Jobs not War," and "U.S. Out of El Salvador." Several American Indians held a banner that read, "We are a Gentle Angry People," and another protester raised a sign reading, "No Noble Causes."

Washington's own Cardozo High School Band, refreshed from its Jan. 1 Rose Bowl Parade performance, high-stepped down the avenue proudly blasting an upbeat disco tune. About 75 youngsters followed them, running along the sidelines. Other units also traveled the route to rousing applause.

Reagan seemed to enjoy the parade as much as those along the 16-block route. Frequently, he stood cherring behind the bulletproof glass in the specially built reviewing stand in front of the White House as rigid lines of military marchers, prancing equestrian units and high-stepping bands passed in review at a majestic two-mile-per-hour cadence.

"It's not so much the man, as it is the event," said Reginald Carter, a doctor from Detroit. "I came here to see the Stevie Wonder concert (following the city's Martin Luther King celebrations) and thought I'd stay on for the inauguration. That's what America is all about."

When the fireworks, the public's finale of inaugural events ended, there were still the nine inaugural balls for Reagan's invited guests to toast what many say is a new era, and dance into the dawn of a new day.