Image quality is tied to the size of the digital sensor and lens...and knowing what the limitations are of those components. A DSLR is going to have the biggest sensor, and any lens can be fitted, including one with a wider aperture (which allows in more light). There are a lot of entry level choices; perhaps the most popular for the money is the Nikon D5100, which can be found for around $500 with the standard 18-55mm "kit" lens. That's a lot of camera for the money, especially when it was released in 2011 at $900.
If you don't want the bulk of a DSLR, you can get a "mirrorless" interchangeable lens camera. Sony's NEX and Samsung NX cameras use the same size sensor as a DSLR. Olympus and Panasonic make Micro-Four-Thirds sensor cameras that overall will be smaller than Sony and Samsung cameras; their sensors aren't as large, but that means the lenses can be more compact as well. Nikon's 1 system cameras are smaller yet.
Lastly, within your budget are fixed lens compacts. Some have larger sensors and wider aperture lenses than typical compact cameras. Cass leaders include the Canon G15, Olympus XZ-2 and Panasonic LX7.
Me and my wife are expecting our second child and we want to buy a good digital camera to save our precious family memories we want to spend around $500. I really don't know anything about cameras but I would like the pictures to look a lot better than my iPhone pics please I could use some help