Describe the parts of the public bath system in ancient rome?

You would go through these is this general order:

Palaestra- exercise room, socialization, preparation for baths basically

Apodyterium- changing room, to get... changed?

Tepidarium- warm bath room, spend time sitting in a warm room. Prepare for the next room

Caldarium- hot bath room, hot water to soak in. After that slaves come rub you down with olive oil (soap was not invented yet)

Frigidarium- cold plunge bath, rinse down the oil with cold water

All hot room were heated by hypocausts, a way of hot air circulating through walls and beneath the floors

tada