2002 Pontiac Bonneville AC Question

The AC (and heater) in my Bonneville has been acting up for a while and I have decided it is time to get it resolved.

Description: The AC and heater do not work immediately upon turning them on. I can feel cold air (when the AC is on) coming out of the vent, but it is very weak even if it is turned to high. After a period of time, sometimes short sometimes much longer like 30-40 minutes, it will kick on and run as normal. Occasionally it seems like hitting a bump or accelerating quickly (getting on the interstate) will cause it to kick on, but this may have just been a coincidence as I know of times where hitting a bump and accelerating have had no effect. Also, turning off the AC after it has started working and turning it back on can sometime cause the issue again.

Any thoughts or suggestions are greatly appreciated.