Sorry for having to be anal about the "car wash", I just wanted to make sure it wasn't really a water related thing!
I haven't had this car for very long, but I was under the impression it might have a stored code but not have the check engine light on. There's no reason for us to be guessing without that info first, just do the 5 pedal stomp thing and see.
Probably not the 02 sensor, the car should run just fine without it. And you would get a code if it were bad.
If the car runs great other than the first few seconds, I wouldn't think dirty injectors, fuel quality related stuff would be the issue either.
My GUESS would be a temp sensor, air or coolant, I don't yet know all the sensors on this car. And again, I would think the ECU would store a code. But the ECU is running in open loop when you first start the car, and if it's getting the wrong temp, it's miscalculating the mixture. Once you get moving, and it starts running fine, that's probably because it's gone to closed loop operation, running off the O2 sensor.
Idle issues are often due to vacuum leaks as well, so get a vacuum gauge and see if it's making what it should. They're cheap.