Hey Jason, don't let these guys get you down. Your project sounds like a lot of fun. I think everyone here needs to go rent a copy of October Sky and just chill out.
I know what it's like to be on a budget, so I'll try to give you some tips on how to do this as cheaply as possible. First off, forget the FM radio. AM radios have been used in r/c gas cars for years, and they are A LOT cheaper. For starters I would recommend the hitec ranger for $39 at
www.towerhobbies.com (stock number LXARP0). If that's still too expensive, you MAY be able to find an older used system on ebay for $20-$30. (It probably isn't worth the effort, though.) The standard servos with this radio will likely be strong enough to actuate your throttle.
I can't give you much advice on the car frame itself, execpt to keep it as small as possible. The smaller it is, the more likely you'll be able to steer it with a standard servo. I picture a motor on wheels. Hey, maybe you could mount suspension right onto the weed eater, he, he. Let us know how it turns out, and of course, use your head...be safe.