Which tires are best for on and off road use?
What you are looking for is a hybrid tire with not enough tread to be a nuisance and slow you down on the road, but enough tred so that when you go off road you are not losing traction on gravel and mud. 

For example Bontrager makes a hybrid tire that is nice:


Also, go with the hard wall tires whenever you can. they cost a little more, but they are more puncture resistant. So they will cause you less grief on the road.