+1 on the drill.
I actually have a corded one and a good extension cord...its sort of a trade off. You can't just pick them up and walk around but corded ones have more power and cost less...in a small apartment with infrequent drill use, I don't mind plugging it in (and unless you keep the batteries topped off, they will have no life when you actually want to use the drill).
Tools that are handy:
Hammer
Pliers (needle and regular)
wire cutters
Screwdrivers (at the minimum, I would suggest TWO with interchangable tips...otherwise one interchangable, one good flat and one good phillips)
allen/hex wrenches - metric seems more common on random furniture but might as well get both
two adjustable wrenches (go one small 6" or so and one big 10-12 and you are covered even if you find something that needs two wrenches).