If you want cheap, a featherweight class drill-box would be a start*. Using drill motors/gearboxes for drive is very cheap, and there's loads of robots out there that use them, or a modified version of. Plenty of those in the build diaries section, check those out if you want inspiration!

As for electronics, a 3/4s Lipo and a set of Botbitz TZ85A speed controllers, along with an Orange R620 receiver (or equivalent 2.4ghz one, if you have one) will get a machine up and driving about, paired with drill motors. Gives a nice base then to add weaponry, if indeed you want to even add any!

Also, reiterating what else has been said, rules are different in the US but it really won't harm to look at the UK rules as well - they're not THAT different but there's a couple of notable changes, like I don't think fuses for LiPo batteries are a requirement in the US, and the removable link can be tool-assisted over there, whereas it can't in the UK. Little things like that I guess.

Good luck with the build!

(*the only drawback is that I believe Robogames in the US don't do featherweights, the lightest they go for the non-insect classes are lightweights, but there's always Motorama and a few other events which do take featherweights...)