Trying to understand jet engine I'm trying to understand at a high level how a jet engine works. I want to know if the following summary I wrote is more or less accurate:
The jet compresses regular air into a combustion chamber. Fuel is sprayed into the combustion chamber. The pressurized fuel/air mixture is ignited and this increases the pressure inside the combustion chamber even more. The hot highly presurized gas blasts out of the combustion chamber. The energy of this blast is captured by turbines. The turbines power the compressor and provide useable power to a shaft.
Is my understanding correct?
 A: With respect, if you want to fully describe the operation of the modern jet turbine engine, you should consider the greatest change in the last 30 years or so,  which has been the development of the large multi-bladed fan at the front of the engine. 
Most modern jet turbine engines are of the high bypass design, with a large fan placed in front of a central core compressor unit. The term high bypass refers to the fact that the front fan is able, by diverting more air around the combustion chamber, to greatly reduce the noise produced by the engine, as well to significantly increase thrust by utilising the fan to  "push" a much larger volume of air against the direction of motion, than a comparable low bypass engine. 
So where is the physics in all of this? The purpose  of the engine to make use of Newtons 3rd law of motion: for every reaction there is an equal and opposite reaction, and the high bypass engine utilises that principle as much as possible by using a set of initial compressor blades, which increase the density of the air by a large factor compared to normal atmospheric pressure, then using kerosene fuel to produce extremely hot, expanded air which is forced through the rear turbine blades,  which are linked by a shaft to the large front fan. Finally, the rear nozzle is carefully designed, using fluid dynamic principles, to produce as much thrust from the hot air as possible. 
I would imagine a significant majority of modern turbine engines produced today are used in commercial aircraft, but obviously the front fan is not suitable for all  applications. 
This cutaway picture shows the main components of a modern commercial aircraft jet turbine engine.
 
A: Yes - although whether it provides "useable power to a shaft" depends on the kind of jet engine (it is possible but not necessary). 
For example, a turbojet does not provide "useable" power to a shaft - it just drives the compressor. A turbofan engine has a bypass path: the compressor does not send all the air to the combustion chamber, but some of it "bypasses" the chamber and is expelled out the back; a turboprop engine drives a propeller, so there you really do get the "useable power to a shaft".
There are some nice descriptions at this site and thousands of others...
Typically a question that has a "yes/no" answer is not a great fit for the format of this site. I tried to expand the answer a little bit beyond what you were asking to make it a better fit...
A: You have the general idea right, but the following statement is subtly wrong

The pressurized fuel/air mixture is ignited and this increases the pressure inside the combustion chamber even more

Unlike in a piston engine, the ignition of the fuel air mixture in a turbine engine increases the mixture's volume while pressure stays relatively constant. Therefore, although the change in pressure across the compressor is equal to the change in pressure across the turbine, there is a larger volume flow through the turbine. This is why the turbine produces enough power to turn the compressor even with losses
This process of compressing a gas, heating it at constant pressure (which causes it to expand), and extracting energy by exhausting it through a turbine known as the Brayton cycle.
The next question is, why do the products of the combustion go out through the turbine than back through the compressor? Consider: the compressor and turbine are approximately the same diameter, and turning at the same angular speed, but because the gas has expanded due to the combustion process, there is more volumetric flow through the turbine than through the compressor. Because of this, the blades of the turbine are angled more steeply to be at the same angle of attack to the local flow. This difference in angle acts as a lower "gear ratio." -- for a given lift, the steeply angled blade generates less axial force and more torque. That's why, for a given pressure in the combustion chamber, the turbine generates more torque and controls which way the whole assembly turns.
