If I create a 8.5x11in file in Photoshop, it works out to 612x792. If I want to create a resume in Illustrator or another program like Sketch by Bohemian Coding should I use that size? Or would it be wiser to use a different one (such as a doubled version of that)?
As long as the proportions are correct and you're not using raster elements, it shouldn't really matter what your pixel size is in terms of getting a quality print. Vector elements are pixel-independent.
Without actually getting in and trying it, I'd suggest keeping 612x792 though, because it'll give you more predictable results when it comes to using fonts at certain sizes. If you double the size, your 10pt text will probably become 20pt text.