For the N1 fan and N2 compressor to move, the turbine needs to spin in an airliners engine. How does energy from the ignited gas spin this turbine? Do the blades on the turbine generate lift just like a wing and spin to spin the N1 fan and N2 compressor?
A turbine is a machine that extracts energy from a moving fluid and converts it into work. In the turbine blades, the air is expanded, which produces work. In many ways, the turbines are the opposite of compressors.
The turbines used in the aircrafts are axial flow turbines, which employ multiple stages to extract work from the air coming out of combustion chamber.
The turbines have two parts- the stationary nozzles (or stators) and the moving rotors. The purpose of the nozzles is to turn the incoming flow so that that it impinges on the rotors at the correct angle, while the actual work is extracted out from fluid by the rotors.
Basically, the rotors are 'turned' by the incoming fluid as shown above which produces work to rotate the shaft (which runs the compressor). In case of turbojet engines (and to some extent turbofan), the air pressure decreases, while the velocity increases. This provides the required thrust.
The main mechanism by which the turbine blades spin is that the flow turns the rotor blades by hitting them, rather than any lift being produced.
I will attempt to answer the question though I'm not an engineer. Each spool (N1, N2 etc) has a compressor in the front and a turbine in the rear. Compressor is a set of rotor blades followed by stator vanes. This is a compression stage. Turbine has the opposite setup: stator vanes followed by rotor blades. They are either pure impulse turbines or impulse/reaction turbines. You may have a look in wikipedia to see the turbine working principle.
Now the high speed combusted air that comes out of the combustion chambers put the turbine in motion. The turbine moves the shaft, the shaft moves the compressor, the compressor feeds the combustion chamber with compressed air and that cycle goes on and on for as long as there is fuel going in the chambers.
Here is a diagram of how is the turbine "set to motion". Red arrows show how the airflow is guided from the nozzles to an impulsive turbine blades.
Image source: own work
The stator vanes are convergent and the air coming through the resulting nozzles is accelerated. As the accelerated air impacts the curved rotor blades, they cause aerodynamic forces on them, similar to those caused on a wing (as others already said). These forces set the turbine to motion.
EDIT: Added image and corrected compressor and turbine stages description