There are a total of 8 different plants to grow to earn this achievement. Once you unlock the Room of Requirement, you can conjure potting stations to grow your own plants.
The plants have different "pot sizing" requirements, however I simply purchased a large potting table from Scrolls and Tomes in Hogsmeade and grew them all there.
Seeds are purchased from several different vendors, however all can be bought in Hogsmeade (north of the main village) at The Magic Neep and Dogweed & Deathcap.
Dittany
Pot: Any size
Waiting time: 10 min
Knotgrass
Pot: Any size
Waiting time: 10 min
Mallowsweet
Pot: Any size
Waiting time: 10 min
Mandrake
Pot: Any size
Waiting time: 10 min
Fluxweed
Pot: Large size
Waiting time: 12 min
Shrivelfig
Pot: Medium or Large size
Waiting time: 12 min
Chinese Chomping Cabbage
Pot: Medium or Large size
Waiting time: 12 min
Venomous Tentacula
Pot: Large size
Waiting time: 15 min
Once you have harvested each type of plant once, your achievement will pop!