Hey Jake,

будьте добры - may sound formal. Usually reserved for slightly formal situations. It can be used to call for attention in a food place (cf. Excuse me...). This is the most formal of all 3, you have to feel the situation to use it properly. In short, it's the most polite of all 3.

Можно мне - can I have, can I get, etc. - i think it's neutral

дайте мне (lit. give me)- if you say it without пожалуйста it is a bit rude (though it's OK in some casual situations when people are not expected to be overly polite) дайте, пожалуйста makes it polite.

degrees of politeness is cultural stuff and it's not easy to perceive them through language only; if you have field experience with these phrases you quickly get the idea.