When you reach level 10 and do the level 10 quest for your class (arcanist) the guildmaster will suggest you can now pick up other classes. That will unlock the "join" quests for all of the other guilds, including crafting.

You can level each of the crafting classes to 15 without leaving the guildhouse if you want - they sell all the basic materials.

In Limsa are the armorer, blacksmith, and culinarian. The others are in the other cities which you can get to once you do the level 15 story quest which unlocks the airship.

As an arcanist you will probably want weaver and goldsmith for gear and alchemist for weapons. These are all in Ul'dah.

To support the crafting class you may want to pick up the gathering classes as well. Weaver mostly relies on Botanist. Goldsmith relies on Miner. Alchemist is kind of all over the place including Fishing.