Does a rocket push up on itself?
No. An object cannot push on itself, meaning that the entire rocket cannot push on the entire rocket.
But part of the rocket can push on another part of the rocket, and that’s exactly what it does. The ship-part of the rocket pushes on the fuel-part of the rocket and the two parts accelerate in opposite directions as a result. The plume of exhaust rushing out of the tail of the rocket is the fuel-part that has accelerated downward to an exceptionally high speed. That fuel-part has been pushed downward hard by the ship-part of the rocket. The ship-part of the rocket has been pushed upward equally hard and it accelerates upward. Gravity introduces a complication, in that it pulls all of the parts downward, but the upward push on the ship-part typically dwarfs gravity and so the ship-part accelerates upward rapidly.