I was in the same boat, I only have 1 bike, so it sorta needs to do everything. I have a Trek 4500 so it came with some super knobby tires, great on dirt but pretty bad on concrete. So i swapped them out for super slicks, which were the opposite, great on concrete, pretty bad on dirt, not so bad on packed dry singletrack actually, but a little mud or incline would ruin the ride. I then tried some Bontrager Connection tires, they are the stock tires on the lower end Trek bikes, they have a Chevron pattern, which is to say, nobbies on the outside and a V shape on thie inside. When they are fully inflated only the centerline V touches the pavement, gives a pretty smooth ride. on dirt they are ok too, but you can also deflate them a little, to allow the side knobbies to dig in. on a road ride i like to inflate them to full pressure, if there is a little dirt i leave them at full pressure, but if i'm expecting miles of singletrack, i drop them to about 40psi each and they have a pretty good amount of traction.
My wife went the other route and bought a Kona Smoke for road rides and a Trek Fuel WSD for dirt rides.