Inspired by The Simple Dollar's 20 Essential Tools, but here I'll comment on what I've found handy for college dorm, car, and home.
Toolbox - this one is required by Man Law. Doesn't have to be huge. Mine is plastic with three tiers that actuall can be taken apart and reassembled. My guess is it should be a foot and a half wide, six to eight inches deep, and probably six inches tall. If your going to be doing a lot of your own work at home or on the car, a tool chest might work, but they're expensive. I stick with the toolbox for mobility.
Tape measure. Measure twice and cut once. Can't do that if you can't measure. I picked one up at the dollar store.
Screwdrivers, dollar store as well. I've never had one break, but it's good to have a set of short stubby ones or extra long of your going to be working on a car. Otherwise, "normal" length will get you by.
I got lucky with needle nose plyers. My female Parental Unit used to work for Sears and snaged me a set of 5. They come in hand for tight places, and while I don't have a wire cutter, they do well in pulling back the insulation when doing some electrical work. Cap'n Barbossa helped me with my programmable thermostat there. :)
Drills - corded or cordless. If your going to do a lot of work, stick with the corded drill. Pink and I usually just need quick drilling tasks, so cordless works best.
Stud Finder - won't help you on a Friday night, but it'll help when hanging stuff. Measure about 5 times to make sure you've got it on the right spot.
Bubble level - *cough* dollar store *cough*
Open end wrenches - get a good set if your going to be working on he car, most house hold stuff can deal with the cheap stuff
Rachet set with both english and metric sockets. This will probably the pricest thing that you buy, but you'll be thankful. I had to have both for my Monte Carlo SS because it had both types of bolts on it. (furthing my complains about why the US hasn't got metric!)
Wire Cutter and striper - this came with my needle nose set, and while it doesn't strip wires, if I'm careful and I can strip wires with some patience.
I have no electrical tester, that's what the Cap'n is for. I do not trust myself around electricity.
Vice grips rule. I have two sizes.
A hammer. Pink's favorite tool. She likes to hit nails.
A step ladder at the least. Pink and I bought an 8 foot ladder I think. We use is to change the light in our car port and my neighbor borrows it to change his. Also helpful when painting rooms and ceilings.
Nails - dollar store.
A shovel, for snow. I haven't had to move any earth yet, but I might be adding some soon.
We have one set, and a hand saw that I used to dear down a 12 foot crab apple tree, trim the rose bushes, and keep the tree limbs at bay.
Lawnmower? Ha Ha! My HOA mows it for me! :P
Lastly, know that the tools will be spread all over the house, so buy extra screwdrivers at the least to save your sanity.