You don't need air to push off of. Rockets, chemical thrusters, and ion engines work in space because they spit out some sort of propellant, spitting out mass in one direction pushes the ship in the opposite direction

nuclear bombs also work in space, you drop nukes behind you and they can push your craft in your desired vector

once you have reached your desired speed, you don't need to do anything to keep going. you WILL need to expend energy or mass to slow down when you reach your destination