Is a compressor blade slowing down the airflow or accelerating it?
are compressor blades actually a barrier to the air flow, or is the airflow accelerated by the compressor


These are not exhaustive alternatives. Keep in mind that subsonic fluid flow when constricted accelerates and pressure decreases (Bernoulli's principle).

What the compressor does is increase the pressure. The speed decreases slightly.


  are compressor blades actually a barrier to the air flow


Actually it's complicated. In a sense they are. The compressor always takes the air at around M0.5 irrespective of flight speed. That means at slower speed the engine needs to suck air from wider stream while at higher speeds some air is spilled over the intake lips. So in a sense, the compressor is a barrier.

But then, the compressor must increase the energy of the stream, otherwise the engine could not produce any thrust. Just the energy goes in pressure, not velocity. So in this sense it is definitely not a barrier.

In fact, the thrust acts mainly on the compressor, because that is the most significant aft-facing surface on which the pressure in the combustion chamber can act.

At supersonic speed it works differently. Now pressure changes can't affect the flow upstream, so the air can't be spilled over the intake lips anymore. Instead now the intake must decelerate it by shock waves and then expansion in a diffusor.

This is needed, because turbomachinery only works well in subsonic flow¹. Since pressure increases as the flow speed decreases, this also increases the overall pressure ratio of the engine and thus its efficiency.

Note that the air also heats up, via adiabatic heating, as it slows down and compresses. Somewhere around M5–6, the air becomes so hot from the compression that adding more heat to it by burning fuel is no longer practical, so above that speed, the combustion must occur in supersonic flow. This is the scramjet engine. It does not have any compressor or turbine, because it gets all the compression from constricting the flow and it turbomachinery wouldn't work at that speed anyway.