There were a whole variety of baths in ancient Rome, ranging from personal baths within a home for use of the household and guests to the massive public baths such as those of Caracalla, which could hold hundreds of people, contained libraries, sporting areas and commons to provide other diversions and were decorated by some of the most famous works of ancient art. The imperial baths were free and open to all, an expression of the generosity of the emperor, and the variety of entertainments on offer arguably make seeing them as just a bath is misleading, they were comprehensive entertainment and relaxation centers that could really fill up an entire day.

However neither of these were really the typical bath--private baths were obviously only available to the wealthy few, and imperial baths represented a true upper end of the spectrum. Between them were baths of all sizes, of all quality and representing all sorts of answers to the question you ask. For example, in Rome itself there were quite a few "neighborhood" baths that served a utilitarian function and charged fees. Some of these would dole out hot water on an individual basis, perhaps for somebody to fill their own tub, perhaps meant to function almost more like a shower (you might poor the hot water over yourself, then dip in the cold pool). There were also communal baths not built by the emperor but rather wealthy patrons or the community itself, and might be subject to targeted generosity--there are examples of patrons donating all the wood needed for heating for a year, or supplying the oil for a month. And there were some baths that were more like health spas (perhaps Gadebridge Park and Bath in England were examples of these), suitable for a trip away from the city.

There is a truly bewildering quantity of scholarship on baths, which reflects their prominence in the archaeological record (they are very easy to recognize) as well as the real importance they held in the daily lives of Romans great and small. Garret Fagen's Bathing in Public in the Roman World is one of the standard entry points.