They're all interchangeable but sometimes the word pretty means that someone is attractive in a cute way. Beautiful and gorgeous means someone is a hot enough to be a model - good face structure, striking eyes and a nice figure - the sort of people that are constantly being stared at.
Hmmm I suppose looking pretty is like the entry level for natural beauty. Most people can look pretty if they look after themselves. But to be beautiful is to be the exception. As for being gorgeous, thats where someone beautiful dresses or behaves sexy and the viewer sees that and will recognise them for being gorgeous? Its all relative though... in short, I'd call a dog pretty and maybe even beautiful, but gorgeousness is reserved for some sexual attraction - and I'm not into dogs, even if they are beautiful haha