Alchemy is taking the Reagents (plants), mixing them with solvents (water) and creating potions or poisons. Plants and water can be found all over ESO and can be picked up for use by crafters within the guild. Crafting experience is know as "Inspiration" and the only way to boost this type of experience is by:
10% for having an ESO Plus Membership
20% “Inspiration Boost” Champion Point Passive (acquired by spending 30 CP in the “Tower” CP tree, so you need to have 90 CP)
To level alchemy, you will need approximately: 22 Natural Water, 43 Pristine Water, 48 Cleansed Water, 60 each of the Filtered and Purified Water, and 20 Cloud Mist. If you have the bonus buffs listed above the amounts will be less. Looking at various bookshelves around the land also allows one to find books on alchemy, which raises inspiration.
Instead of re-writing books of information, the following are some excellent links to aid in crafting potions and poisons:
Tari responds, "Magic? Pfft! I got me a bear!"