No. Rockets don't need to push against anything to propel themselves.
Don't feel bad if you got this wrong. We heard one of the stars of "Mythbusters" flub this concept. He calls himself an expert, but this makes us wonder -- an expert at what? He was trying to propel a boat with jets of compressed air. It wasn't working too well so he said he was going to point the jet's blasts toward the water so they would have something to push against.
This seems to make sense. But for 500 years we have known it's wrong, thanks to Isaac Newton's discovery of a principle of nature called Third Law of Motion. This describes the fact of nature that for every action there is an equal and opposite reaction. You've experienced this if you have every fired a gun and felt a recoil. The bullet is the action and the push against your shoulder, actually the gun trying to go the opposite direction, is the reaction.
But it doesnt' have anything to do with guns. Exert a force in one direction and there's an opposite force in the other direction. In the case of the rocket, the gas exploding out the back is the action, like the bullet, and the movement of the rocket in the opposite direction is the reaction.
Its a good thing rockets don't need to push against somthin' because space is basically filled with nothin' and space exploration would be impossible. Then we'd have to recall all those satellites and spacecraft we've sent up for having succeeded under false pretenses.