Ignore BFG, he's a little grumpy from being up so early.
Sounds like what you are looking for is a cyclocross style bike. It is set up like a road bike (for the most part) with a stiff frame and wheels for the offroading portion that you are looking for. Problem that you are going to run into is that there isn't much, if anything, in your price range. My suggestion is that you save up a little more and really research so you make a really informed purchase. Ride lots of bikes so you find something you are going to love.
I love noodles.