The main noise you hear is the starter motor. This is a high torque motor whose purpose is to turn the engine at a rate sufficient to allow it to run.
If you run a starter motor with no load it is still quite noisy, but as you will hear when starting your car, it tries to get the engine up to speed, fuel is pumped in, the spark plugs fire, and if all is well the engine spins up. Otherwise you have to try again.
Because the cylinders are still compressing a fuel and air mix, the starter motor does not simply accelerate, but instead sounds quite variable as it passes points of high and low compression.
The load is so high that car manuals recommend taking a break after every few tries to avoid overheating the motor, and it is very easy to quickly drain a battery using the starter.