If the friction on an object is 200 N and you apply 200 N, it seems, the forces should immediately cancel each other out since the friction and your applied force go in opposite directions, leaving you with an object that is motionless. Yet, from what I've read, that's not the case. I feel like I'm misunderstanding something. Could it be I'm wrong about needing to apply 200 N initially? Might the amount of force I actually need be greater, but instantaneously decrease to 200 N?
Forces are only affecting acceleration. Not any other parts of motion. Think of Newton's 2nd law:
If the sum of forces is zero as in your case, nothing accelerates. If it was standing still, then it continues standing still. If it was moving then it continues moving!
It only takes a force to change a motion. Not to keep it up.
Answering your Title question. NO, the objects do not change their state of motion if all the forces cancel each other. I will assume that the friction on the object is maximum. That means $200N$ is the maximum friction on the surface.
Now the force needed is $200N$ to overcome the static frictional force. So if initially you apply $200N$ force, there will not be any motion.
But the scene changes if you are applying the $200N$ force to the moving object whose maximum kinetic frictional force (suppose $190N$) will be smaller than than maximum static frictional force (i.e. $200N$). Yes, the body will experience smaller friction when in motion. In that case the object will be supplied constant net Force of $10N$. So the velocity will also change due to the acceleration.
I think you're failing to distinguish between static friction and kinetic friction (the latter also called dynamic friction), e.g.,
Briefly, once moving, it takes less force to keep an object moving against friction than it took to get that object moving in the first place.