So here's the basic idea -- In FFXIV, if you want to play as a certain class, you equip the weapon/tool of that class. If you want to be a Carpenter, you equip a saw and you're a Carpenter. If you want to be a Marauder, you equip an axe and you're a Marauder. If you want to be a Miner, you equip a pickaxe and you're a Miner.

In order to complete "Fieldcraft" leves, you have to buy and equip the appropriate fieldcraft tool. In order to complete "Battlecraft" leves, you have to buy and equip a weapon. And in order to complete local (crafting) leves, you have to buy and equip the tool for that particular craft (a needle, a saw, a hammer, etc.)

Check out that Yellow Gremlin site to figure out where to buy "weathered" (i.e., rank 1) tools and weapons in your city. Then, buy a weapon, pick up some tutorial leves, go to the appropriate camp, equip the weapon, and run those leves.
Problem is, due to a lack of information, I have no idea how to even get to the abilities menu, whether I even have this Arbor Call ability, etc.
Open your main menu, and go to "Actions and Abilities." After you've equipped a botanist's tool, you'll be able to select Botanist abilities to place on your action bar.

Welcome to the game, and good luck!