You'll need the basics to start. Sockets and ratchets, pliers, dikes, screwdrivers, wrenches. Start with that stuff. I'd recommend Craftsman for someone starting out, if only because you'll learn what you like (and don't like) in a tool and you can upgrade later on. You will find that as you progress you'll want the higher quality tools more and more. Beyond the basics, just keep a running wish list of stuff you find out that you want. EBay and garage sales are also good options for finding used tools cheap.
Now, here are some tools that are must-haves (in case you get a tool set that doesn't have them) when working on airplanes:
1" wrench. It's the size you use for removing oil filters.
7/8" deep socket. That's what you use for spark plugs.
And a few of my personal favorite tools in my box:
Matco ratcheting screwdriver. Second most used tool in my box.
2D LED Maglite. You can never have too much light when working on and inspecting an airplane, and the LED Maglite is indestructible. This is my most-used tool.
Safety wire pliers from Matco. They just make life so much easier.
I have a running wishlist too. Number One on there right now is a good (think Fluke-level) digital multimeter. I love doing electrical work and a good meter is worth its weight in gold for that stuff.