The Alchemist mixes herbs in order to generate potions with a variety of effects. A player can create healing, invisibility, elemental resistance, and mana potions; oils to coat weapons; and much more.
Alchemy is useful to all character classes. Melee fighters can use alchemy to create buff and healing potions. Spell casting classes can use alchemy to create mana potions and related buffs. Alchemy is a fun Profession where you can create potions for yourself, sell them to others, or give them to your party members and friends. Alchemists are well loved when they hand out potions.
First, you need to find an alchemy trainer. You may have to ask other people where to find a trainer. Alchemy trainers are usually located near an herbalist trainer.
You don't need any special equipment or devices beyond the required reagents. Alchemy always requires flasks to make potions. You can make potions anywhere.
If you wish to be an Alchemist you should already be an Herbalist. It is possible to be an alchemist without being an herbalist if you buy or are given the required ingredients by other players. However, it's not wise to be an alchemist if you aren't an herbalist.
Potions are created from herbs, like enchantments. You will also need to buy various reagents from the Profession merchants as well as find rare reagents from monsters, fishing, chests, or other players.
The Alchemy trainer will teach you many recipes. Not all recipes can be taught by a trainer. Additional recipes will have to be found from monster drops, chests, or (more commonly) purchased from other players. Make sure you tell your friends you are an Alchemist and they may bring you recipes and reagents.