They are for the most part a very reliable car. Just find one that's been reasonably well taken care of (kinda goes for any car though). No real common bugs with an '03.... Some of the earlier cars, 1999-2001 had issues with the coolant crossover section of the intake manifold cracking but that was remedied in the later half of the 2001 model year.
My bro in law has an '03 GT auto he bought about 3 years ago. He does nothing but drive it, put gas in it and change the oil occasionally. Notice I didn't mention "wash it" lol. Only problem he's ever had with it was a heater hose blew out one time. He got the car at 107K, think it has around 150K now.
I've had my car for going on 8 years, it's at 116K currently. Only actual repair work I've had to do is replace the clutch. And the alternator, but that was the fault of some crappy mod I did and not the alternator's. Had a misfire issue a few years ago, new coils and plugs fixed that.
All around pretty solid cars. Not super fast, in case you're looking to piss off LS1 people, but solid and fun cars.