Lily Herman gets very specific when it comes to proofreading. In an article for The Muse, she becomes the Terminator of English errors, which is a lot less exciting than being a regular Terminator. I guess it might be more practical though.
She says to start by overviewing the most basic spelling and grammar. (Google it if you need help with the grammar.) Next, make sure all proper nouns are spelled correctly, which is something local newspaper printers could not do if their lives depended on it. All verb tenses need to agree too.
Then you move on to overall sentence structure, identifying sentences that are too long and unwieldy and breaking them up into more digestible parts. After that, you look at formatting issues like margins. Additionally, be checking for consistency, to make sure you do not use both “%” and “percent” for example. Another often forgotten biggie to check for is whether you are using idioms correctly; “could care less” and “couldn’t care less” mean different things. Finally, you just have to read it to see if the whole thing flows naturally. Getting someone else to read it should be the fastest indicator of whether it makes good sense.
For more tips, you can read the full article here: https://www.themuse.com/advice/8-things-to-check-when-proofreading-anything