Rockets have allowed us to launch robots, animals, and people into orbits around the Earth — and even beyond. They do not work by "pushing against the air," instead, rockets use momentum or the amount of power a moving object has.
Rockets work by ejecting hot exhaust that functions similarly to a basketball. The gas molecules in the exhaust do not weigh much, but they escape the rocket's nozzle quickly, giving them a lot of velocity. As a result, with the same overall oomph, the rocket moves in the opposite direction of the exhaust.