I think the easiest way to explain this is, 'unto' is a word that doesn't get much use anymore outside of scripture and other Old English literature. It means 'to." So anything you might use to for, such as, "Bring it to me," would work with unto—"Bring it unto me." It just makes it sound a lot more formal and no one really uses it.

Onto indicates more of a where something/someone might go, like "We climbed onto the roof." So, if you're putting something on top of something else, that's when it's most appropriately used. I would say to use it sparingly, though, because the word on by itself may be all you need. For example, "I put the collar on the dog" is sufficient, though you could say, "I put the collar onto the dog." It just sounds like the collar could be laying on top of the dog rather than around it's neck, like on would infer.