Alchemy is the big name for health potion and mana potion training. There were ideas of an alchemy class but that got lost in the void of time. Both are trained in Falconreach.
To get to the places, you just start at Falconreach and go 1 screen to the left.
You can train your alchemy level up to your lvl +1
So if you are lvl 20, you can train your alchemy up to lvl 21.
The maximum level of alchemy is 40 so you have to be lvl 39 in order to maximize your alchemy.
Training your alchemy level for MP and HP potions increases the amount it heals. At lvl 40 it can be very helpful to heal 625 HP instead of 100 from the beginning ^^