Friends & I are going to do the same thing.
These are the rules for our local derby. Though, yours will probably be different. It's run annually every year, last weekend of August.
http://www.saru.ca/sitebuildercontent/sitebuilderfiles/2012_derby_rules.pdf
They won't allow door bars here (Which I don't like), there's also no welding of bumpers (Another dumb rule IMO). But eh, if you don't like the rules run somewhere else I suppose. Few things you will need to buy will be a boat gas tank, and some metal for the crossbar behind the seats & grater blades for the doorbars.. If you want you can cut the roof sign out of metal too, or wood. And find some old pipe for the headers if you want to run 'em out the hood. For your case though you probably don't car. Just whatever works for you. I know we're like that right now. First time thing & all.
Car wise, depend if they have classes or not. E.G. compacts/4 cylinder, full-size, pickups ect. Obviously compacts are going to be the easiest to obtain, and probably the cheapest - It won't be as much fun as the full-size V8's.! Prices will probably be between $100 - $700 depending on the car, again. Obviously any pre-1978 Chrysler, pre-'79 Fords and pre-'77 GM cars will be the best. Any '77+ GM car
will bend down in the rear (where most of the hitting is done, 'ya know?). That is, unless you pre-bend the frame, cut pieces out of the frame or add leaf springs. A lot more work unfortunately. '80s Fords such as Crown Vics are good, and '80s Dodges like the Diplomat, Gran Fury ect. are good.
There's more, maybe I'll write some more later. Or let you ask any questions you have after reading this.

👍
EDIT: Nothing better to do right now, so I'll continue...
Once the car is purchased (You won't need the papers for it, so if there's a car without regee then it's no big deal.) Gut it, remove all trim, hubcaps,
everything. drop the tank, install the new one, weld up the bumpers, trunk, doors (Or chain them if you have no welder). Bolt on the doorbars or fill the door(s) with cement. Weld bars or steel mesh across the windshield. Cut holes in the trunk for inspection, cut holes in hood for fires and/or headers for the engine. etcetera. Until all the requirements are fulfilled. Afterwards the paint, roof sign and all the fun stuff can be made & painted up.
So, cars done! Anything else? Yes sir! -Many derbies have entry fees, our particular one in Salmon Arm's $90. A car trailer will be required for carrying the car + any welders, tools, spare parts, tires and radiators you bring along. So make sure someone has a trailer + vehicle that can town all the weight. 👍
Pretty much the quick overview off of the top of my head. Anymore questions just ask.
Also; Here's one of my favourite Youtube channels. They're all about derbyin' & there's a lot to learn from them!
http://www.youtube.com/user/demolitionmissions?feature=results_main