Best camera under $500

For a light and semi-pocketable carry around camera you could check out a refurbished E-PL5 from Olympus, you can get the camera with a 14-42 lens really cheap and get a viewfinder for it. It doesn't do 1080p@60fps.

The 1080p@60 really cuts out a huge number of cameras though. Also, lenses aren't generally rated in Yx Zoom when you are looking at normal camera gear, it's more of a marketing term. The reason is that something like a 7-28 would be a 4x lens, but also a 28 - 112 since the focal length ends at 4 times where it begins for the zoom range.

If you are looking for something that zoom in pretty well, be looking for something that gets in the 100-200mm range if you are going to be hand holding the camera. It's much more difficult to get a steady picture at something like 500mm or 800mm while being hand held unless you get really lucky on the image stabilization and have a huge amount of light available. The further you get in to telephoto the more it amplifies your small hand movements.

Plus, you have to think you are going to be carrying it around all the time. Plus using it for images inside too, so maybe you really want something more like a Canon S95 or Olympus XZ-2 that has a fairly good zoom range but also a pretty wide aperture number (A lower f number on the lens) so that you don't have to crank up the ISO as much when you have less light.

One thing I usually tell people to avoid is these ultra-zoom style cameras because there is usually too much of a trade off somewhere to get the lens to be able to do all things it just doesn't do anything really well.