I would like the footer's background color to be a different color than the rest of the website. Is there a proper way of doing this or do I just create a rectangle element, stretch it out onto the footer area and change its background color?
There's no "proper way" of doing this, but there are many ways to accomplish the same result.
You can go as simple as just creating a rectangle and stretching to the size you want for a quick-and-dirty solution.
If you plan to reuse that footer you can turn it into a component and maybe make it Auto Layout, for example.