What crafting class should I pick up first?

[Niwashi]

If you want to start with just one, Culinarian is the most independent of the others, as Frizze discussed. Its only downside is that it tends to need a lot more ingredients, so if you're taking the independent route of gathering your own mats, you'll spend more time on that than on actually crafting.

If you want to start with a couple of them, Weaver and Leatherworker complement each other well since they use a lot of each other's materials, and starting there has the advantage that you can make most of your crafting/gathering gear. I find this a good starting point. (And making my own crafting gear tends to make these two, particularly weaver, pull a bit ahead of the others even when I try to balance leveling all of them.)

Once you've tried it out enough on a class or two to decide whether you like it or not, if you decide to continue further, you will end up wanting all of them. A good initial goal for expanding to all eight is to get all of them up to level 15 for the first set of cross-class skills, and Culinarian up to level 37 for a particularly good cross-class skill. (After that, the next good skills come at level 50, which takes a bit longer to reach.)