Really, you're free to start whenever you want. It can certainly break up the level grinding and story if you have other things to do. As a crafter, it will be really good for you if you're able to gather mats as well. For culinarian, I would highly recommend leveling botany (and having a ton of inventory space because that craft has a zillion ingredients.) For armorsmithing, mining will be really helpful to you. Plus you'll need shards to craft things, which also come from botany/mining.

So basically, give them all a try. There's no rush, but feel free to start whenever.