Why don't rockets need air to stay up, like planes do?

Part of a rocket's propulsion system includes a chemical oxidizer, either in the form of liquid oxygen, the solid ammonium perchlorate, or the propulsion/oxidizer mixture Hydrazine. No outside source of oxygen is needed. 

Steering is accomplished by "gimbaled" rocket nozzles which change the direction of the blast to keep the rocket on course. Once in space and in orbit, slight changes in orientation is made by small rockets placed around the craft that spurt little gas jets to nudge the craft into the desired position.