What crafting class should I pick up first?

[Niwashi]

You can get most of the non-crafted materials for Culanary from Botany, but Mining provides the various shards that all crafts use.

To clarify this just a little: You can get all the shards from either Botany or Mining. However, at level 20 you get cross-class skills to increase your shard yield. If you have both BTN and MIN to at least that level, then you can use those skills for all six elements. With just one or the other, you'll only have access to three of them (lightning, fire, and water from Mining; earth, ice, and wind from Botany).