You could bathe at home, if you were rich, but you would miss out on a lot of social life if you did. Actually, it was at an earlier time that Romans had private baths (lavatrinae) in the house. The public bath was an idea borrowed, like so many others, from the Greeks. Although it was the mother of the emperor Augustus who started the fashion for the nobility of attending public bath houses, in truth and in fact everyone went.
There were different kinds of public baths, but in general they were well-lighted, with large windows, and became more and more elegant. Many were free, for the poor, but the charge at most baths was only the smallest coin (although women paid twice as much as men). Some had separate chambers for men and women, some had separate hours, and there were some baths where everyone bathed together. This was not considered respectable, even though people wore garments like swimsuits (unless they and the establishment were REALLY of questionable character). Women could be attended in the bath by male slaves and massaged by them afterwards.
Generally, you went to the baths after the midday meal and siesta around 2 or 3 p.m. So Trimalchio and his friends are going much later (just before dinner), trying no doubt to sober up.
When you went to the baths, first you entered a waiting room. Next there was a changing room, with shelves. Then you entered the tepidarium, or warm room, which was also equipped with shelves to hold towels, oil, and the strigil. After the warm room, logically, you went into the hot room (caldarium), which was kept at a lower temperature for the women. In the caldarium, there was a tank of water for bathing and a fountain for your hands and face. First you soaked, then you were rubbed with oil (by yourself or a slave), and then the oil and dirt were scraped off with a strigil. (There was no soap.) Then you went back to the tepidarium, and after that to the cold plunge (frigidarium). Then you toweled off and returned through the tepidarium to the changing room.
But there was much more to the baths than simply getting clean, or even keeping cool in the Mediterranean climate. Men's baths had places to exercise, and women's baths had beauty parlors. There were cafés nearby for both sexes. The public baths were luxurious recreation centers, including gymnasia, libraries, gardens, art galleries, covered walks, terraces, and shops. Under the emperor Nero, there were nearly 1,000 baths in Rome. Seneca lived over a small bath house and complained of the noise.