Quick answer
You do not “go to the sauna”, you идёшь в ба́ню. The steam room is пари́лка, and the greeting afterwards is С лёгким па́ром!
The place
The kit
What you do
The etiquette
Three rounds is normal: steam, cold plunge, rest with tea — then repeat. You sit on a sheet, not on the bare wood. The felt hat is not a joke: it protects your head from the heat. And nobody throws water on the stones without asking the others: Поддать?
С лёгким па́ром
Literally “with light steam”. It is said to someone coming out of the banya or the shower, and there is no English equivalent — the nearest is “hope that felt good”. Answering Спаси́бо! is all that is required.
Typical mistakes
Я иду́ в са́уну — meaning a banya
Я иду́ в ба́ню.
A са́уна is the dry Finnish kind; the Russian one is wet.
Sitting directly on the bench
Sit on your sheet
Both hygiene and a burnt skin issue.
Check yourself
What do you say to someone leaving the banya?