What are some of the most common mistakes authors make when they edit themselves? What should we be watching for as we read through our work?
All authors have their own pet words and phrases that they use much to often. They are so familiar to you that you can't see them when you read your own work. Keep a list of those words and do a find/change or find/delete before you submit.
Same with –ly words. I'm not one of those who think they should never be used but most of the time you can reword it more strongly. (ha)
As a writer, you're describing what you see and hear in your head. You already know what is going to happen and how. As a reader, they will be taking their cues from your words. They won't know the scene until you paint it for them. You know your character and story so well that you miss the gaps. So will people who know you, like family and close friends. This is why you need other people to read your manuscript.
Spelling and grammar. If you consistently misspell a word, or make a particular grammar error, you may not realize it's wrong. Use your spell checker and grammar checker, if your software has one. (They may not always be correct, but chances are they'll be right more often than wrong.)