To become a Botanist, you'll need to first reach level 10 with your primary class and, if you did not start in Gridania (it's the forest area where the Archers, Conjurers and Lancers start), progress enough until you can visit other cities in the airship.
Once you are able to, visit the Botanist Guild in New Gridania and you'll be given the first class quest, upon doing it, you'll earn this trophy.