I have a 2001 Pontiac Firebird. When i start my car, my left turn signal light comes on but the right front turn signal light remains off. If i turn my lights on, the right turn signal light then comes on also. However, even when my lights are on and both turn signal lights are on, only the left turn signal will flash, the right turn signal will just stay on.

Perhaps a dicky connection somewhere. Perhaps corrosion, even moisture in the connection plugs. Inspect all connections to your indicator lights. Also check for worn or damaged wires. Also Have a look inside the lenses of the lights to see if the the globes are corroded or have water ingress etc and causing shorts to other wiring within the lighting circuits.

You may also consider turn indicator lever to on to indicate truniong right, make sure the engine off. Have someone watch right indicator and literally grab the plugs/leads to the indicator globe and give them a shake to see if the fault goes away. If it doe stop intermitantly then it will help point you in the correct area.