This image that TomMcW posted in the comments is a much better representation of what is going on.
First, looking at axial velocity only is only half of the story, through the compressor and especially through the turbine the airflow can be more circumferential than axial. In fact the point of the stator vanes and turbine nozzles (turbine vanes) is to turn the airflow to give the proper angle of attack on the blades.
The velocity drops in the compressor because the flow area doesn't decrease as quickly as the air density increases, by design. This increases the efficiency of the compressor and keeps the air moving slower at the compressor exit.
The flow area increases a lot at the compressor discharge going into combustor, this is to slow the air down for better combustion and to ensure the static pressure on the outside of the liner is higher than on the inside. In the combustor, two things happen to increase the velocity, first a massive amount of heat is added, decreasing the density requiring a higher velocity and the area gets smaller, also increasing velocity, both can be calculated with the continuity equation. You also get a small amount of mass increase through the combustor from the fuel.
Now at the exit of the combustor you have the Stage 1 turbine nozzles, which accelerate the airflow to Mach 1, and generally with a very sharp circumferential angle. The air then slows and is straighten as the turbine blade extracts work. This is repeated through each stage of the turbine, although it will not achieve Mach 1 again.
Also remember Mach 1 changes throughout the engine with total temperature.