It's more about mood than anything else.
to gaze -> kind of a dreamy quality - like gazing at a beautiful sunset or at the person you love
to stare -> a bit harder, more intense - staring at a person is a bit rude, for example. You might also stare at something you're trying to understand, like a difficult diagram.
to peer -> usually used when someone is searching for something - like peering into an alleyway to see if anyone's there
Hope this helps!