Begin with crafting and gathering

[Niwashi]

As for the intermediate materials issue, some are more dependent on each other than others are. If, for instance, someone were planning on leveling weaver without leatherworker or vice versa, it'd be worth pointing out that those two are quite tightly linked with using each other's mats. Culinarian, on the other hand, is almost completely independent of the others, hardly ever using their materials and I don't think ever providing mats for them. (At least, if there's any exceptions I haven't come across them. I'm level 50 with all eight.)

It used to be essential to level CUL at least to 37 for Steady Hands II, but now that that one has been moved to having each DOH class learn it, the only reason to level CUL now is making food (including food-based housing items).

ALC is kind of in between. It does provide some materials needed by other classes, so you'll need it if you want to be fully self sufficient. But its mats are used less often cross-class than those of the classes you've already picked to advance, so buying the ALC materials needed for the others might work for you.

ARM and BSM have some overlap, as they use the same types of metals. For instance, right at the start, both ARM and BSM can make bronze ingots, but only ARM can make bronze plates. It may be worth picking up at least one of the two (both if you want to be self sufficient).

I'd recommend unlocking all eight (or seven if you want to leave out CUL). Then focus most of your attention on leveling the ones you've picked, but when you have the time, do all of their GC daily turn-ins. That will gradually get you some levels on the others with minimal effort, giving you a bit of a boost if you later decide to go back and level them as well.