How is an elevator kept in constant velocity if theres no net force acting on it? I get that forces are what accelerates an object but if the tension force was equal to the force of gravity and the object was already in motion, would it really just move up? Hard to picture. 
 A: That's exactly how Newton's first law works. No net force = no acceleration. Note that "no net force" means that the cable has a tension equal to the force of gravity - and moving that cable under tension does require work to be done. So the elevator doesn't move "for free".
Think about a puck sliding on an air hockey table. Almost no net force, it glides along... 
A: The answer to your questions is ..... Yes it will go up with uniform velocity if the tension force is equal to the force of gravity given that it already has an upward velocity. As the lift gradually rises it increases the tension force slowly and then decreases it to the previous value (that it had when it was at rest). Yes force of gravity changes with height but that little change is little enough to be neglected. And after that just like @Floris said its Newton's first law. When both the forces are equal there is no force regulating it's velocity and hence its value does not change. It rises up with a constant velocity.
A: Even if there is no friction, we have to keep pushing on the elevator to stop if from falling down, pulled by gravity. But if it were far out in empty space, away from all gravity, the elevator would keep moving with constant speed in a stright line when we stop pushing.
What keeps the elevator moving is inertia. This is difficult to picture because it doesn't happen in real life, only in ideal situations such as empty space. In real life everything slows down and stops when we don't keep pushing it. That is because there is always some other small force like friction or air resistance which is present whether we push or not.
