The "camera" won't make the difference, the photographer will first and foremost and the glass in the lens a close second.
The difference between 12 - 16 mp will be insignificant for an amateur shooting. Generally, the more mp, the more the pic may be enlarged and printed before you see pixelization. More mp also helps with cropping and preserving the ability to print larger sizes.
The difference between an okay photo and a great photo can also be in the post processing with software such as Lightroom or Photoshop. To get the most benefits from such software, you need to shoot RAW images and not JPEGS. The G16 will shoot both RAW and JPEG allowing you to get familiar with shooting in general and before you go to the expense of purchasing such software but I would recommend ultimately getting one or both (they are not inexpensive however - they will cost more than the camera). There are cheaper alternatives but you get what you pay for.
If you are going to shoot only once or twice per year, don't bother with anything over $500. The G16 is at that price point, for example, and will shoot HD video in addition to stills. 12mp should suit you fine with your limited intentions.
As for zoom capability, make sure whatever camera you choose is based upon "optical zoom" and not "digital zoom". Digital zoom, IMO, is worthless and destroys photos taken by amateurs who think more zoom is better automatically when digital will cause pixelization and optical will not. Get the most optical zoom and completely ignore any touts of digital zoom.