Richard and Lorraine West stood speechless yesterday morning in a crowded national airport lobby as a ticket agent calmly described "A Little Tour" they would have to take to reach their home in Salt Lake City.

"You'll see the Manhattan Skyline, then the Golden Gate Bridge and the beautiful Pacific coast." said agent R. Edwin Melvin. "That's the good news."

But for the Wests and thousands of other airline travelers yesterday there was little good news. United Airlines, the Nation's largest airline, cancelled all of its 1,600 daily flights as its machinists went on strike for higher wages.

At National Airport and elsewhere, the strike produced frantic searches by travelers for alternate flights. Telephone lines to other carriers were jammed and many rushed to area airports yesterday, hoping to find a seat, any seat, headed toward their destination.

"I'll take any place within walking distance," one man shouted at a United agent yesterday at National Airport. "But just don't send me to Boise."

The Wests, who had planned to fly back to Salt Lake City on a nonstop four-hour flight, were among the lucky ones. Their homeward trip, which United will pay for, will take two days, including on overnight stay in San Francisco, and several closely-timed cross-city taxi rides to catch connecting flights.

"Oh my God," wailed Lorraine West, "I'll never see my luggage again."

United Airlines accounts for 23 percent of the nation's domestic airline business, according to a spokesman. The strikes is costing the company $10 million a day and all its flights have been canceled through April 9.

Last night, a member of the National Mediation Board announced that the striking International Association of Machinists, which walked out at midnight Friday, had agreed to meet with United negotiators early this week, probably on Monday or Tuesday, in Washington.

"How long will it last?" said United customer service supervisor Nick La Rosa, as he slumped in his chair at National Airport. "Tomorrow would be too long."

At National, where approximately one-fourth of United's 1,500 employes are on strike, nonunion personnel yesterday soothed the stranded.

"We've had a few screamers," said ticket agent Elaine McDonald, "but for the most part, people have been pretty nice."

United normally handles 5,100 Washington area passengers each day, with 66 daily flights originating from National, Dulles and Baltimore-Washington airports.

"A lot of people didn't even know we were on strike," said reservation agent Pam Santos, standing behind the congested ticket counter in her stocking feet. "Some people came up to me this morning complaining that there was no one checking in bags," she said. "Any they weren't foreigners either."

When Holly Zanville arrived at the airport yesterday morning, and saw that her Denver Flight was canceled, she let out an expletive.

"I'm stuck," she tearfully told the ticket agent. "How about Minneapolis?" he offered. Zanville opted for another night in Washington.

A man from Idaho next to her was passing out plastic lapel pins in the shape of a potato. Other travelers lugged suitcases throught the smoke-filled lobby, fanned themselves with read crumpled newspapers and United airlines tickets.

"My husband's going to kill me", said Doris Stewart, who was trying to get home to Boise, Idaho. "I was staying at the Howard Johnson Motel, but they said they have no rooms tonight. The closest friend I have are in New Jersey. I haven't seen them for 20 years, which should be interesting."

Before her, a ticket agent was reeling off instructions like a carnival barker. "Go to TWA or American and head for Chicago," he directed one couple. "They'll probably drop us in Detroit," muttered the couple, sipping beer from paper cups.

On the National Airport sidewalk, two machinists set up a picket line, reminiscent of the 1975 strike which shut down United Airlines for 16 days."That was hectic, but this is worse," said ticket agent Elaine McDonald. "We're all chained to the computer now."

McDonald said the ticketing of United passengers to other airlines was being done by telephone because United's computer could not handle the overload.