Self-Editing is a great book.
When I edit (either my own work or other's), I look for these things:
- extraneous adjectives, adverbs, descriptions, etc.
- run-on sentences or wordiness. Most writers overwrite and everything can be pared down if you pay attention.
- Grammatical issues: for example, you have no idea how many seasoned writers still misuse participial clauses or semicolons.
- Cliches: most are placeholders during first draft, and that's fine. But during edit, try to find other ways to say what you need to say. Something more original?
- Dialogue: I find that more often than not the dialogue is good enough to move the plot along, etc. but they can always be tightened or improved. I always find my dialogue 1000x better after a few edits.
- Logical flaws: that's something much harder to do and you may need another reader, or a beta, to help you spot those problems. My betas are great in that they tell me things I didn't even think about. When something bothers them, chances are there is something wrong with the plot/character/story, etc.
- Structure: I look for structural problems, and those include prose- and story-level issues. Sometimes moving a paragraph or chapter to another place makes things so much better. I recently did just that with my opening chapter and I was very happy with the result.
- Too much internal and not enough external. Too many I's. To much "he did this/he did that." To much navel gazing.
- Flow and pacing: story flow problems are difficult to spot. Again, it's good to have another pair of eyes to help you with this. When your eyes glaze over something or when you're bored or you want to skip forward, chances are something is wrong with the pacing or the story flow.
There are other things (including typos, grammar, etc.) you will only improve with practice. But after reading the text a millionth time, it becomes harder and harder to spot those niggles.