Question

If an object were released in space far away from planets or stars and given an initial momentum (mv), describe what would happen to that object.

Answer 1

Would just keep going with the initial momentum? There is no air resistance and if it is far enough away from planets or stars than gravity really would affect it so it couldn't be pulled in to a planet/star or even orbit them. I'm pretty sure that's what would happen. 

Answer 2

This is a perfect time to review Newton's first law of motion:

"A body remains in uniform motion until acted on by an external force."

Uniform motion means zero acceleration, and THAT means
constant speed in a straight line.

The way you described your object, there's no external force
acting on it.  So it continues in uniform motion ... straight line,
constant speed, constant momentum, zero acceleration.