I can't answer for the Greeks, but the Romans were more concerned about hygiene than one might think. They really, really liked to take baths. The types of baths and how people bathed changed over time, as this funny letter from Seneca shows, but the Romans always really loved being clean. Many richer people had their own private baths in their houses, but for the poorer folks they built huge public bath buildings from at least 100 BC onwards. Many Roman towns outside of Italy had their own bathing facilities too.

Many later authors (so AD not BC) mention upper-class people bathing everyday, or even multiple times a day, in the afternoon. I don't know if we can assume everybody bathed daily or in the afternoon, but I think it's safe to say most of them bathed regularly at least. Many people would also wash at least their arms before eating.

I don't think that I am at all qualified to say how the baths worked or how they were customarily arranged. I do know that a lot of the bathing customs varied according to time and place. In Italy, for instance, Pliny says that men and women would bathe together, but this practice was outlawed several times from Hadrian onwards and we know that this was never popular in many Eastern parts of the empire. I think I can only make a very basic outline of the "Roman bath experience": First, you would put oil on yourself. Then you would go through a variable number of hot, cold, or tepid baths. At the end, either you or a slave would use a thing called a stirgil to scrape the oil along with all your dirt and sweat off. Sometimes the oil would be scented, but not always.

Speaking of scented things, the Romans also regularly used perfume, primitive shampoo, and a kind of scented tooth paste. as Catullus tells us, some people also tried to use animal pee to whiten their teeth. Which, to their credit, would work; but it wouldn't smell very good at all.