I forgot one thing.
Yes, pressurization usually comes from “bleed air” from the engines. (turbocharged piston engined airplanes get their pressurized air from the turbochargers)
What is bleed air? Well a turbine engine operates on pretty much the same principle as any other internal combustion engine. You know that old saying suck, squeeze, bang, blow. (aka, intake, compression, combustion, exhaust)
The only difference is that it all happens at once.
A turbine engine is as close to a perpetual motion machine as you can get.
Step one. Suck.
This part is easy. During start, the starter motor (or pressurized air) starts the whole system spinning. Once the engine gets going, it happens automatically as the first part of the compressor section. See step two.
Step two. squeeze.
The first stages of the engine are the compressor turbines. These turbines accelerate the air and then squeeze the air with a centrifugal compressor. This is where the bleed air comes from. Even before the fuel is added the air has been compressed so much and hot enough to be used for pressurization and heating of anti-ice devices (leading edge wing heat, boots etc.) Hell, it’s even used as part of the engine fuel system.
Step three. Bang.
During start (and during other times as neccessary like heavy rain/ice/turbulence) the electrical ignitors provide the spark to ingite the fuel. Once running, the fuel ignites automatically as a continuous flame.
Step four. Blow.
The exhaust gasses acclerate through another series of turbines, finally providing the power to make the airplane “go”. Either through turning the propellors or exhausting out the rear to provide thrust. These turbines also provide the rotational energy to the first half of the system. Remember those first compressor turbines? They get their power to do all that compressing from the back half of the turbine, from the expanding exhaust gasses. Cool, huh?