Hola!
Well, it depends a bit on how familiar you are with your interlocutor and the degree of friendship/comradeship with each person. I don't think there's a general rule for using the three of them, as it may even depend on your mood upon saying goodbye at a certain moment: you might just say "un beso" to your girlfriend (which would sound pretty deadpan) to finish a chat session because of being upset :/...
Anyway, let's say we'd use "un beso"/"besos" in a more formal way; a "besote" sounds like the type of kiss to someone with whom you have a close friendship but not intending to go 'beyond" and finally the "besazo" kiss would definitely go to someone who you are specially interested in (u know what I mean :D), someone who you want to kiss in fact and not because of mere formality...
This said, be careful since we might use the three kisses between relatives as synonyms, or between just friends (to the opposite sex) in letters or emails, to be more or less affectionate!
I hope you don't find this a mess...Spanish is sometimes weird :)