When a rocket fires its engines, it is “pushing” against the gases that exit. Newton’s 3rd law, which you seem to cite, states that every action has an equal and opposite reaction. Think of this as forces being mutual. Multiply a force by the amount of time that force is applied, and you’ve got the change in momentum. So a rocket “pushes” its exhaust gases out very quickly out the back, applying force over some time. Those exhaust gases have momentum in that backwards direction. The universe conserves momentum, however, so the rocket has to have equal and opposite momentum (so the momenta cancel each other out). This moves the rocket in the forward direction. Since the rocket is more massive than the exhaust gases, the rocket will move more slowly but have the same momentum (which is equal to the product of mass and velocity).
The force doesn’t have to push against anything. For example, gravity pulls us towards the earth without pushing against anything else. Strictly speaking, the thrust from a rocket engine is pushing against the rocket.
We tend to think that we need to push against something because that’s how our bodies work when we walk or swim, but a sturdy balloon deflating in a vacuum would fly around just as easily as one flying around in the air (more easily, in fact, because there’s no wind resistance)
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