Turbine engines come in a variety of designs that range from a single shaft to which most parts are connected, to complex designs involving multiple shafts and stages that work independently from one another. How the starter turns them depends on the particular engine. In all cases, however, the part of the engine in which the burning takes place and the parts attached to it, may be thought of as the "core," sometimes referred to as the gas turbine generator, or just simply as the gas turbine. It's this that the starter turns.
The names N1, N2, N3, etc, become somewhat confusing. The easiest thing to remember is that the heart of the engine is the part with the fire, where the first stage turbines are located (the first turbine wheels downstream from the burner cans), and which drive the highest stage compressors...the ones closest to the burner cans and the last stage or stages through which the air passes before arriving at the burner cans. This "core" or "gas generator" can be thought of as the engine that drives the engine...everything else is being driven by the core.
In the high-bypass turbofans, with a very large frontal section and big fan blades (as you see on most airline equipment), the actual core to the engine, the N2 section in a two-stage engine and the N3 in a three stage engine, is much smaller than the engine you see hanging on the airpalne. Most of what you see is what's being driven by the small core. I like to think of the core as the poodle, and what you see on the wing is the poodle before it gets wet or gets groomed. It's just a very expensive, very powerful poodle.
The starter is connected to the core through various means of gearing. The starter motor is actually attached outside the engine, usually to a shaft in the accessory section. that shaft works through the accessory section gearbox (or in some cases directly) to turn the core gas generator. It's this action that starts sucking air in at the front of the engine, and provides just enough airflow to allow the engine to light off at the right time and accelerate. The other stages of the engine don't really contribute much at this point in time...they're available to do a lot of work at higher power settings, but aren't doing a lot for you in the starting process.
The compressor sections of the other stages are turned by their own turbine wheels, located downstream from the core turbine wheels. The core takes energy out of the gas flow coming from the burner cans and uses it to turn it's own compressor discs or wheels. The other stages get what's left of that energy to turn their own wheels...and it's slow going at first. Often when starting, the N1 stage may not even be seen to rotate for 15 or 20 seconds...the core needs to build up some speed first and until light off, it's only the air being drawn into the engine by the core compressor that provide any motive force to turn the other engine stages.
The first parts of the engine to turn, then are the accessory section gears (or tower shaft, which is the part some starters attach to, to drive the engine), the core turbine, and the core compressor. You can picture the core section like an axle on your car, with a wheel at each end. Start turning the shaft with the starter motor, and you have the wheels turning at each end. One wheel, aft of the burner can, is the turbine wheel that gets turned by exhaust gasses, the other wheel is the compressor, ahead of the burner can.
The question was asked regarding back pressure. As the compressor begins to build up pressure, it dumps it into a small chamber expands the volume, and slows the airflow down somewhat. This is called the diffuser, and where the airflow decreases in velocity, it experiences a pressure rise. The goal of the compressor is two-fold; draw in as much air as possible, and boost it to the highest pressure possible. That's all good and well as long as the air can be kept moving. But at slow engine speeds, the air isn't moving very fast at all...and suddenly it gets backed up at the diffuser.
Ever get off a flight and find that half a gazillion of you want to fit through the door at baggage claim? Moving toward the exit works great so long as you've got a lot of room, but when you get to that door, the line slows up. There's pushing and shoving, and the pressure builds...much like in the engine. If only many people can fit through the door, the ripple effect moves aft, and people begin to slow down behind. You see the same thing as cars get backed up on a highway. Airflow is getting sucked in as fast as the engine can make it go, but can only move through the engine so fast.
The pressure rise in the middle of the engine means that the pressure at the front of the engine is less...and just like people who don't like to get crushed in the rush for the exit, that airflow desires to move to the area of lowest pressure. It doesn't move forward so well, seeks somewhere to go, and that seeking action is called back pressure.
To relieve that back pressure, "bleed valves" are installed which allow a certain amount of pressure to automatically leak overboard, out the sidesof the engine, until the engine is turning fast enough. Some bleeds open at certain speeds, some due to demans inside the engine, but the effect is the same. These bleeds are often referred to as acceleration bleeds, and what they do is bleed off excess pressure to allow the engine to accelerate, and keep airflow going through the engine.
Another way to think about it is an "unloading" process. The inability to move air through the engine and the reduction in velocity as it approaches the diffuser result in increasing resistance to airflow (back pressure), and an additional load on the compressor (and in turn the turbine wheel, and as we're talking starts here, the starter motor, too. Bleeding off some of that pressure and excess air unloads that resistance, allowing the engine to accelerate more efficiently. Sort of like spillgates in a dam getting rid of extra water.
The blades in the compressor work much like propeller blades or any other airfoil, but inside a very small room, or duct. To be most efficient, the airflow meeting those blades has to be at a narrow range of angles. When the airflow starts backing up, the "angle of attack" or angle at which the airflow through the compressor meets the blades can quickly become an inefficient angle, or even a stalled angle...this is where the term you've probably heard comes from...a "compressor stall." Acceleration bleeds are placed at various points in the engine to prevent this from happening, especially during start.
As the engine turns faster and faster and can process more of that airflow, the acceleration bleeds close and allow all the air to pass through the engine.