I like the cyclocross idea, just because the sport is cool. There are a couple of other options, though, particularly if money's tight.
One is an old (mid-80s or so) steel mountain bike, like a Hard Rock or Bridgestone. You can get them in thrift shops here, in good shape, for $50 or so. Tires and a tuneup, if you can't do it yourself, would run maybe another $100. They're decent to ride with road tires (I did several 50-60 milers on my Bridgestone) and pretty durable.
Another would be a touring bike of about the same age. They're road-ish bikes but usually with relaxed geometry and room for fat tires. Harder to find than mountain bikes and probably two or three times as expensive, but if you find a good one, it will be sweet.