G: Are you going out for exercise?

M: No, I¡¯m going to the library.
G: Oh, really?
Can I ask you to pick up a book for me?
--------------------------------
What expressions can we use instead of 'pick up'? Can we use 'check out' or 'borrow'?

Yes, either "check out" or "borrow" will work fine. You could also use "get" here.