Mountain bikes have meatier tires - wider and knobier, which give more traction. Also, it helps to have a fluid pedal stroke. Mashing will make your tires slip more.

As for the front end rising it because your position is too upright. first, try lowering your elbows (parallel to ground) and scooting up on your saddle. If you're still popping wheelies, slam that stem or use barends - or both