You can certainly switch to a more appropriate tire. Schwalbe Big Apples are a popular choice -- I use their 26x2.0 tires on my mountain bike. They roll well on asphalt, and are quite functional on packed dirt and gravel. They're not suitable for loose dirt/gravel, rocks, roots, etc, you'll need a knobby mountain bike tire for those.
The only real downside to this plan: If you intend to ride your bike in the city, **and** often ride trails as well, you'll quickly tire (pun intended) of switching between your mountain tires and your city tires. If you reach that point, you might consider buying a separate set of wheels and keeping the mountain tires mounted on one and the city tires on the other.