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