When some of us were kids, we'd get corrected if we announced to our mothers or teachers a sentence along the lines of: "Me and her are going snake-hunting in the creek." "It's 'she and I,'" they'd say, apparently more concerned about proper grammar than the state of our shoes after the excursion.
But they were right: when the pronouns are the subject of the sentence, we need to use the subject forms: I, we, you, she, he and they.
Conventions of English dictate that you don't start a pair with "I," but it's not grammatically incorrect to say "I and my cousins went bungee-jumping in New Zealand." (It does sound a bit odd, though.)
But we tend to run into problems with object constructions. We get so conditioned to say "you and I" that we want to use it everywhere, as in: "Just between you and I, his feet smell terrible." However, "between" is a preposition, so we need to use the object forms: me, us, you, him, her and them. So "between you and me" is correct. The same goes for "Doodle's going with Cindy and me on the snake hunt."
Between you and me, a quick way to determine the correct word is to replace the pair of pronouns with "we" or "us." If "we" sounds right, use the subject forms. If "us" sounds right, use the object forms.
"[Me and her] -> us are going snake-hunting in the creek" Nope. Use the subject pronouns here: She and I.
"Just between [you and I] -> we, his feet smell terrible." Nope. Use the object pronouns here: you and me.
And watch out for snakes in the creek.