The president and Mrs. Reagan began a marathon of ball-going last night with a visit to the American Legion's banquet in a capital celebrating both the inauguration and the hostages' freedom.

"What did he say?" asked a latecomer in the Washington Hilton's Grand Ballroom, where 6,000 Republicans were squeezed into a space that might have been comfortable for 3,000, with a dance floor adequate for about 100 couples.

"He said he was delighted to be here," said a friend who had been present when President Reagan arrived, precisely on schedule at 9:15, and spoke for five minutes.

Actually, the president said more than that. He drew cheers with an announcement that the hostages had landed in Algiers and were ready to switch to American planes, and more cheers when he said: "I just won't call them hostages; they were prisoners of war." There was also loud applause when he announced that "I have finally decided, I'm not going to wake up if this is a dream."

One guest who was not cheering was Evelyn Y. Davis, editor of Highlights and Lowlights, a Washington newsletter, who has been to 40 parties since the inaugural weekend began and seldom has to buy a ticket for any political function. "I paid 100 bucks to come here," she said, "and I walk in and find a cash bar. I'm a non-drinker and wouldn't mind paying for a drink if I wanted one, but the idea of a cash bar at something like this turns me off. It's cheap."

Some others were unhappy about being where they were. "We paid $2,000 for a box at the Kennedy Center," said California businessman Ray Jensen, "and we checked by phone to see that they had our reservations, but when we got to Washington our tickets weren't there and we end up standing here instead of having a box at the Kennedy Center. I wonder where you go for a refund."

Still others seemed totally content. "I'm happy to be a part of this," said Lance Torrance of Athens, Tex. "I hope he scores a couple of quick touchdowns, and then we'll really be rolling."

Virginia Steiger of Los Angeles was a volunteer helping with Reagan's national correspondence during the campaign, and she watched him come and go with stars in her eyes. "I got a personal letter from him to thank me for volunteering," she said. "I am very enthusiastic about the guy and proud to be an American."

Most of the guests stayed in the Grand Ballroom, where one observer estimated there were at least $2.5 million worth of women's clothes, and listened to Woody Herman purveying nostalgia with such tunes as "Satin Doll" and "Don't Get Around Much Anymore."

By 11 p.m. when Glen Campbell was performing, it became physically impossible to fit any more people into the Grand Ballroom. Uniformed firemen stood at the doors holding back large and angry crowds of ticket holders. "I just went out to the men's room," one man was saying, "and now I can't get back to my friends on the inside."