Does the velocity of air in an engine increase in the compressor section or turbine section?
Air velocity increases in the turbine section. 

In order to compress air, we need to apply work on it, like running a car engine harder to drive it up the hill. Part of the work required comes from decelerating the air that flows into the engine.

The work done can be released by accelerating the gas and letting it escape, creating thrust. The expanding gas also applies work on the turbine blades, which absorb part of the accumulated energy. Each turbine stage slows down the gas, but from a higher velocity base due to the expansion. In the turbine section as a whole, gas velocity is increased. 

The faster the gas flows out of the engine, the higher the thrust that is created.