A good general toolkit should include:
Hammer - don't get a big framing hammer, just a medium claw hammer will work
Screwdrivers - at least 2 each of flat and Phillips
Ratchet set - Handy for any car work you might want to do, and for assembling furniture, playsets, etc. I'd get a 1/4" handle with metric and SAE bits.
Wrenches - Handy as ratchets, but for tighter areas.
Torpedo level - You want your pictures to be straight, right?
Drill - You can go cordless if you'd like, but if it only sees occasional use I'd get an inexpensive corded drill. You'll have more power, and won't have to worry about keeping the battery charged. I bought a low-end Dewalt for about $60 and it does most of what I need. A drill is a good general-purpose tool that every house should have.
Tape measure - At least 12 feet... 25 is better. Wider ones cost more, but you can stretch them out unsupported for longer distances.
Pliers - at least one each of normal and needle-nose
Vise Grips - Not always needed, but when you need them, nothing else will quite work.
Soft-sided tool bag - You want to keep your tools more or less in one place, so that you can just grab the bag and go to where you're working. I like the soft bags, but that's me.
When you use up your current toothbrush, clean the paste remnants off of it and throw it in your tool kit. Little brushes can be awful handy.
For the hand tools, I have mostly Craftsman stuff. Great warranty, and a known name. You'll pay more than if you go to WalMart and buy cheapies, but you are getting better quality. What I did for a while to build up my "general purpose tool" collection after I moved out on my own was to go to Sears on every payday and spend about $20 on one of the above. I also bought tools as-needed if I ran into something that required one I didn't have. After doing this for a while, I now have a respectable "homeowner" level tool bag, and can do some car repairs.
Another handy thing (but at a higher "level" than hanging pictures) is a small inexpensive multimeter. Radio Shack has a little folding yellow one for about $20 that is well worth it. I used mine (along with a little "How to Use your Meter" book) to troubleshoot the phone plug in my bedroom when I moved into a new place and didn't have to get the phone company to come out and fix it. When my wife felt a shock while using the stove, I got my meter out and verified that if you touched a pot on one eye of our stove along with a pot on any of the other three eyes (assuming they were on), there was current flow.
That the kind of stuff you're looking for?
