Is it normal for a bike to skid even after both brakes are applied? When I apply both brakes going at moderate speed, it takes a few feet before the bike stops and I can hear the rear tire skidding. 

The brake pad seems to be working fine since the wheels themselves quickly stop rotating, it's just that the bike continues to skid even when the tires are locked in place. 

Is this normal behavior, and what can I do to improve the power/responsiveness of my brakes? Might it be a tire rubber issue?
To answer the question shortly: yes, it is normal.

When braking, the inertia of the rider (and the bike) causes the load on the rear tire to decrease. It leads to lower friction against the surface, and possibly skidding. In extreme cases, when braking with the front brake, the rear tire is lifed off the ground completely.

Especially in fast mountain biking, the rider must lean back or "hang" as far back as feasible when braking, to maintain better traction and prevent possible falling over the handlebar.