I searched frantically in the wiki and tutorials on a way to train my dwarves; but it seems to hint like I assumed; that unskilled dwarves are very ineficient, but still capable of carrying any tasks.
Is this a bug? Or did I miss the "academy workshop"? I'd really like to be able to train my dwarves from scratch.
This is intentional. You can, if you want, take a dwarf who has been mining his entire life and never set foot aboveground and for absolutely no reason other than because you feel like it, tell him to go pick berries on the surface. He'll suck at it at first, and it'll usually take him several bushes to get even a single usable plant, but he'll be able to do it. As he continues doing it his Herbalist skill will gradually increase, and he'll become better at it, getting more plants in less time.
To get your dwarves to build a mechanics workshop, turn off all labors on one of them, turn on the mechanics labor, and designate where you want the workshop built. If you've already set a workshop to be built, cancel that order and set it again.