It's not pushing against anything. It's expelling mass (rocket exhaust) in one direction and Newton's third law (conservation of linear momentum) says that the rocket has to move in the opposite direction. Exactly the same thing is responsible for the "recoil" or "kick" when you fire a rifle. It's not because the bullet is "pushing" on anything, it's because the bullet left the rifle barrel at high speed and the rifle moves in the opposite direction with the same momentum as the bullet (which, because the rifle is much heavier than the bullet, is a lot slower in velocity terms).
You’re half right. Newton’s third law applies to the entire system.
However, from the reference frame of a rocket, the expelled mass exerts a force on the rocket engine that must be reacted through the engine to the engine mounts, and up through the primary structure.