I'm coming over from a Nokia Lumia 920 & 928, and I have to say that while I LOVE the G2, I'm not super happy with the camera either. It takes a good photo, but that's it. The Lumia took a SUPER photo, almost every time, no matter lighting, movement, etc...
This camera seems to be finicky and does the auto focus too much. Low light is fair to good and moving photos are not good at all. If I use the Intelligent Mode, it is much better, but rarely does a person have the time to start the camera app, change modes, then take the shot. Also, the Intelligent Mode is super slow to take a shot. By that time, your subject is GONE. The Lumia was less than 3 seconds from pocket to photo, and it came out great.
I'm not knocking this phone, or camera, again, I LOVE the G2, but the hype about the camera is not warranted. If you want to see what I'm talking about, go into a store and try the Lumia 920, 928, or 1020. You will immediately see the difference.
I'm hoping a Firmware update will fix the issues. The Lumia's went through a period of needed Firmware updates, then all their camera issues were fixed, maybe the same will hold true here.