What does the rocket ship push off of in order to change its velocity?
A rocket ship pushes off of gas expelled from the engine.
In short, A rocket ship pushes off of gas expelled from the engine.
Motion in a total vacuum with no influences is determined by Newton's Third Law of Motion.
Using this law scientists have determined that
So when the gas weighs 1g and is moving 10m/s and the mass of the rocket is 1g, the rocket must move 10m/s.
Motion in space is not as simple as
Thrust for rockets are (typically) by RCS thrust, SAS thrust, and main engine thrust.