Pretty good.
With my instructor head on just for a second (I try not to bring it home with me), I would just say this:
Air is air. It only becomes "Bleed air" when it is taken from a compressor for some other use.
So our little Auxiliary Power Unit at the back of the aircraft normally has a valve on its compressor which when open allows air to "bleed" away from the APU along a network of pipes known as the pneumatic system where it can be used for many different tasks.
Examples might be:
Air conditioning,
Pressurisation of the cabin,
Pressurisation of the water systems,
Toilet flush!
Anti-ice systems (though not usually as the demand is too high)
and engine starting.
For engine starting, the air goes through a valve called the start valve to an air driven starter motor known often as an Air Turbine Starter motor. Because of course it is a motor which uses a turbine to convert pressure (of the bleed air) into mechanical rotation. This ATS is connected to the gearbox and therefore turns not only the gearbox and all the other pumps and things connected to the gearbox but also as you said the engine itself.
On your picture the yellow shaft is connected to the red "spool" or "rotor" and this will then rotate. When this spins the compressor on the front of that rotor starts to draw air through the engine from the inlet to the exhaust.
The picture you posted is a "twin spool" or "dual rotor" engine so the air entering the engine flows past the "Fan" and low pressure compressor shown in blue before passing through the high pressure compressor then through the combustion section, out through the high pressure turbine (which is doing no work at this stage) the low pressure turbine (which is extracting energy to help turn the fan) and out the back.
When the two rotors are turning fast enough (often this is dictated only by the speed of the red high pressure rotor) we can add fuel and a spark. The combustion of that fuel with a small % of the air going through the engine adds pressure at the two turbine sections, this makes the engine accelerate, as the engine accelerates the amount of air (mass) passing into the combustion section increases and we can add more fuel which in turn creates more pressure.
This continues until the energy from combustion alone is enough to keep the engine turning. This is known as "self sustaining" speed. At this point we no longer need the help of the ATS and the start valve can close.
But the yellow drive shaft is still connected so now the engine is providing the power to drive the gear box and all connected pumps and things. We don't need the ATM to turn anymore so some kind of clutch or freewheel device is incorporated to allow it to stop. This is important as the ATS is one of the fastest rotating devices on the aircraft. 50,000rpm on some models and we don't want them turning when not needed.
Hope that helps a bit. I would also suggest looking at many different engine types and aircraft too as you will find many variations in design and names for things but the general principal is roughly the same.
Bonne Chance young man!