basic ideas:
1) conservation of momentum. momentum = M mass x V velocity
hot, pressurized, burnt fuel goes back and rocket and unused fuel goes forward. the process is continuous so the calculation can get messy so assume the burn is a short time and all the fuel mass is used up at the same v
2) gravity force that is the weight of the rocket body and fuel is pulling the rocket down
3) Force F = MA or more correctly F = change in momentum/ time of change.
4) units have to be correct to make sense. Pound is force, not mass. Kilogram is mass not force. Newton is the unit of force in the metric system.
5) kinetic energy ( after the fuel is burned and the rocket is coasting) = 1/2 M V^2
Staging is away to throw away the big heavy empty tank and heavy powerful rocket motors after they have been used up. What is left, a smaller rocket, lighter fuel and smaller engines is moving with a high speed and is already high where the air drag is less (or none). It is then easier to gain more speed to reach orbital velocity.
It is fairly simple if you study basic physics.
The Chinese were using rockets 1500 yrs ago. But they did not understand the F=MA physics.
Think of it this way. In a gun the powder explodes and the bullet goes one way and the gun the other.
Bullet small and fast, gun big and slow
Rocket burn: no bullet, just the mass of the fuel very fast one way. the rocket (like the gun) goes the other way.
PS the rocket fuel exhaust does NOT Push against the air. It pushes against the bell shaped rocket motor exhaust chamber