I don't really have a lot to say about editing for grammar. If you want to do it, you're probably going to have to read up on how. It's the basic sort of thing that you just need to know, or you can have just about anyone do it for you. You don't have to be a professional, but at the very least the average writer should have some grasp of the language they're using to write.
If you want to know your grammar really well EDIT YOUR OWN WRITING AND GIVE CRITIQUES TO OTHERS. If you just stare at it and say to yourself (I see the error), you are more likely to forget how to correct it than if you take the time to change it.
Likewise, if you find the error in someone else's story, take a moment to [politely] correct them, possibly even explain it, and you'll be reminding yourself, too! This is a learning technique, and it really does work.
Part of being able to, say, recognize a Granny Smith apple turnover from a Macintosh one, comes from being around the two. Once you know what to look for, it really isn't any harder than picking out the orange in a barrel of apples.
There are some basics that you should know, and those I'll mention here.
Don't worry about buying a fancy book. Often, the best thing you can do is go to google and type in what you want and read a few of the top results. With variety, you'll learn pretty quick what's correct and what isn't.
I've found it's easy to learn one step at a time based on what I need. If I need help with commas, I only search for commas. If on Monday I need to brush up on my semicolons, I search for semicolons on Monday.
THE BASICS
(IE: consider googling these terms if you want to make your life easier):
Independent clause vs dependent clause
Commas
Commas and conjunctions
Comma splices
Semicolons
Hyphens and Dashes
Gerunds
Making sure your modifiers modify the right thing
Prepositions
Punctuation in dialogue
Ellipses
Colon
Whose/who/whom
Affect vs effect
Lay vs lie
STUFF I'M ALWAYS FIXING:
This sentence is over. Capitalize the first letter of the next one.
"How do I punctuate a question in dialogue?" Bruce asked.
"I don't know how," Sally said.
RUN ON SENTENCES. I CAN'T EVEN.
WHEN TO EDIT FOR GRAMMAR:
All the time. As you go. As the words are coming out of your head and appearing on the paper, screen, used napkin, whatever it is you're writing on.
Seriously, don't make more work for yourself later. Punctuate. Use your commas. Don't say "Ah, I'll go back in later." Make it a habit and sooner or later using grammar won't even be a conscious decision!
Finally, if you ever need help, please feel free to comment or PM me and I will be happy to assist you or direct you to some fantastic websites that might help!