I am very confused about this. I know how to start a job, but I can't figure out how to assign a dwarf to it. I've looked around a lot, and can't find it. Can anyone help?
You've got it backwards; Dwarves don't pick jobs, jobs pick dwarves. When a job is created, it picks the nearest dwarf with the appropriate labors enabled.
If you've set a job and are finding it's not getting completed, chances are, you don't have enough dwarves with the corresponding labor enabled.
While you can assign labors manually (Magmawiki link), this method usually becomes untenable once you have more than a handful of dwarves.
Therefore, I would suggest you look into Dwarf Therapist, an external program that you can run alongside dwarf fortress which will take care of all labor assignations from one simple screen.