OK, I'll give it a go:
-if it's an older bike with a threaded head set, you'll need some headset spanners (which will probably have to have a thin profile, so they are a bit of a specialist tool). Otherwise if it's a threadless headset, it's usually just allen key to undo it.
-if you want to remove the headset races, you will need a specialist tool (but it's dirt easy to make your own, just google "home made headset removal tool")
-chain -you'll need a chain link extractor (but don't forget to check to see if you have a removable link first!).
-depending on what your bike has, you'll need a special freewheel or cassette removal tool. No getting around that. On the bright side, you can make your own chain whip quite easily with an old chain (again google).
-for the crank, you'll probably need a crank extractor -again a specialist tool. Some people advocate just riding without retaining bolts until it loosens up and you can take it off, but I just don't like the idea of that (and that's not to say it doesn't work either).
-for the bottom bracket, again you'll need a specialist tool.
I assume you're happy with not dismantling the wheels and hubs..... Anything else you can remove with what I call "regular" tools -e.g. screwdriver with different bits, set of allen keys, adjustable spanner, etc. It's also really handy to have a big adjustable spanner and cable cutters too (just makes it easier I think). Pretty much anything but the cassette/freewheel, headset and crank and bottom bracket don't require any special kind of tools.
Anyway, I really recommend you do this; it will teach you how to take stuff apart -and of course you can always reassemble as well. If you do buy tools, buy quality; they'll make the job easier and will last you a lifetime. You can end up doing a lot of stuff your bike shop used to..... good luck!
p.s. Oh and don't forget the plastic thingmajig cable guide on the bottom of your bottom bracket -when you need one, you'll need one and it's handy to have a spare. Usually very easy to remove with an allen key!
p.p.s. if you are like me, I cut up the frame and use the tubes for extra leverage for mechanicking.