I have a 99 Passat GLX VR6, auto. I just bought the car a couple of weeks ago.

In the morning or if the car has been sitting cold for 5-6 hours, after the first start I notice that the right front part of the engine compartment is making a whirring sort of sound. If you put the car in gear and drive, you hear this vibrating sound like something is loose. The car seems to not want to shift. Once you have driven for about a minute the whirring sound goes away, the vibrations stops and the car is a smooth as can be.

I noticed also that if you let the car idle for 1-3 minutes at startup you can hear the whirring sound and then all of a sudden you hear something click and the whirring sound is gone. Everything is smooth running at this point, no vibratiing noise.

Any ideas are appreciated!

Do you have any accessories running? Perhaps you have the fan on in defrost position which will cause the A/C compressor to be on, loading the engine while it's cold, causing a vibration. When the compressor cycles (clicks) off it idles better and the whirring noise quits. It's just a guess.