During approach to PIT CL-65 Captain experiences a TCAS RA on a general aviation aircraft; a go-around due to traffic on the runway; and a GPWS warning on the missed approach. All of these events are attributed to PIT Approach and local controllers.

Narrative

On approach into PIT Approach descended us into a conflict with a single engine aircraft. We received a Resolution Advisory from TCAS. First Officer and I both had visual contact with the traffic and agreed there was no imminent threat to our aircraft. We did not deviate from our heading or altitude. On final approach to 28R into PIT with the aircraft configured for landing and inside the final approach fix. PIT Tower cleared an air carrier flight that was holding short of 28R for takeoff. With that aircraft on the runway we executed a go-around. While on the go-around from 28R in PIT we were given a right turn to 360. While in the climb and turn we received a GPWS warning. 'Terrain Terrain Pull up Pull up.' We executed the escape maneuver. The controllers in PIT were having a bad day. The only thing I can suggest is the controllers pay attention to their jobs. There is no way the Tower Controller was looking out the cab window when he cleared the air carrier for takeoff. He would have seen us on final. As for the Approach Controller; extra training on working with general aviation and 121 aircraft.