OK, these are pretty different cameras.
The Canon G11 is a compact camera. It is very easy to use, and convenient with its small sensor and attached lens. The smaller sensor however will struggle in low light.
The Panasonic G1 is a little different. It was one of the pioneering micro 4/3rds cameras. This means that you will need to buy a lens separately to use it. It also has a larger sensor which will give more detail in low light, but in bright daytime conditions, image quality will be much the same, as in fact most cameras are.
As the G1 is an interchangeable lens camera, you can buy specialist lenses for it for your particular needs, for example telephoto lenses, or wide angles, or bright prime lenses for shooting in low light.
Both these cameras are a few years old though. Digital technology changes fast. I really would recommend buying the latest technology available at the time (in digital) even if this means you have to step down a camera or two. Ie if you cant afford the latest Canon G12 or Pano G3, then Id recommend something like the Canon S95.
Personally Im a big fan of the Fujifilm F550
Nikon D5000 and D300 with 12-24mm, 35mm f1.8, 50mm f1.4, 60mm f2.8 Macro, 18-200mm, 17-55mm f2.8 (all Nikon)