If I start my car up then takeoff right away I hear a clicking gurgling sound when I accelerate. The car stops doing it after a few minutes and it wont do it at all if I start the car up and let it idle for a couple minutes before taking off.

It's a 94 Z28 (LT1), auto trans.

Is it my oil pump?

 

Maybe it's related to the air pump which runs for a little while when you first start the engine. Maybe poke around the hoses that come from the check valves on the exhaust manifolds and make sure they aren't burned through. Let us know what ya find.

 

To follow up on Kevin's post, open the hood, take the cover off the fuse/relay box, start the engine. While you hear the noise, pull relay E out and see if the noise stops. That's the relay for the AIR pump. Alternatively, you could pull fuse #7 in that box, same result.

It will set a code, and your fans will come on, but put the relay or fuse back in and everything will return to normal. The stored code will clear itself eventually.