Hey Ivo what's up buddy I am not familiar with other aircrafts but I am enrolled in a Learjet 24 class so I will go ahead and describe you how the starting system works on the 24.
The Learjet 24 is fitted with two General Electric CJ610 rated at 2950 lbs of thrust at 102.2% of N1. To start the engine we use the starter generators, serven both as starter and as generators. The main controls for the starting procedure are the Bttery switch, the GEN OFF START switch and the throttles, in addition we monitor the N1, the ITT and the FF gauges. When you place the battery switch to ON, a circuit is completed to energize a solenoid operated relay that closes allowing the current to flow to a battery bus. When you move the GEN OFF START switch to the start position the stby electrical fuel pump energizes; the fuel motive flow valve closes (under normal circumstances motive flow from the main fuel controller is used to drive a jet pump that pumps the fuel from the tip tank to the main supply line); and ignition system arms. When the motive flow valve closes a circuit is completed to energize a solenoid operated start relay allowing the current to reach the starter wiring of the starter-generator. As the armature electrifies and becomes and electromagnet itself, as a reaction to the magnetic field generated by the stationary coil, it starts to spin transmitting the torque to the gear box and then to the compressor to which is connected by means of a tower shaft. As the engine reaches 10%N1, as indicated on your tachometer, advancing the throttle past the FSO Fuel Shut Off position into the Ground Idle one, the exciter box is energized powering the two spark plugs and the fuel is introduced into the combustion chamber. As the fuel air mixture ignites and expands, it generates a stream of hot gases blasting through the turbine making the compressor assisted by the starter to accelerate to IDLE equal to about 48%N1 at sea level. Upon reaching the Idle the GEN OFF START switch is set to GEN, the start relay opens, the generator control relay is energized cutting off the flow of current to the armature and the starter generator reverts to its generator function completing acircuit for the voltage regulator that keeps the output of each generator at 28.5VDC, a circuit for the voltage equalizer bus to allow to evenly divide the strain between the two generators, and a circuit for the freon compressor of the air conditioning system. Finally the fuel motive flow control valve opens, the stby pump is isolated and the ignition system is disengaged. As the engine accelerates the output of the generator increases gradually up to 28.5VDC where it is metered by the voltage regulator. During the start up procedure you want to make sure that you have a battery voltage of at least 22VDC in order to prevent a hot start or a hung one and that you have the inverter switched on cos the oil pressure indicator is AC fed.
Other commuter sized turboprop or turbojets have similar starting system with a starter generator assembly.
Hope this helped