look for cyclocross tires to get something with a bit of grip. I used them for winter riding on a plain old Trek years ago.

As far as dirt roads go, as long as they aren't very loose or very steep, you probably don't particularly need knobbies for them. My husband and I did a little riding on forest service roads (which around here are mostly hard packed dirt with a bit of gravel) with our old sport/tourers with their regular tires. The biggest problem was probably being careful of not puncturing a tire on a sharp rock. Some tread will at least keep the littler rocks from damaging the tires. Even now I ride with an unrepentant crosser and it seems like at least 1/2 the rides have some short bit of off roading that even my race bike can handle....