Well, you can only take 4 crafting leves at a time. My understanding is that if you pick up 4, complete them, and turn them in, you can pick up another 4 right after that.

But yes, leves and grinding are the two methods to level up crafting, at the moment. When you join a Grand Company, you'll also be able to do supply quests for your GC to earn GC points.

And don't forget about Guild Tasks. The Adventurer's Guild in every city has a "Task Board." The Task Board offers two simple tasks for every gathering/crafting guild in that city. The two tasks are pretty low level--around a level 1 synth, and a level 10 synth. In addition to a reward of gil/shards/crystals, when you turn the required materials into the NPC at the guild, the lower level synth offers 300 bonus XP, and the higher level synth offers 500 bonus XP. You can do one a day for every guild. And unless they've changed it, you get the XP for whatever class you have currently equipped. So, for example, if the Carpenter's Guild Task for the day is to create an oak lumber. You create the lumber, then switch to Weaver, and turn it in, and you'll get 300-500 XP for your Weaver class.

It's not a ton of XP, but it adds up. 500 free XP a day = 3500 XP a week = 15,000 XP a month.