I agree with Tino, in this instance, me is correct. But me and I present a huge problem for English speakers. And it does not seem to go away! So forgive me for giving a very long answer. Me is an object pronoun and I is a subject pronoun (meaning when action is performed on "me" I am the object of that action so "me" is used, but when "I" perform action the subject of the sentence is me so "I" is used). Okay, that was a little cute. A "trick" that may help: "This question is between you and me." Delete the other pronoun and preposition and see what sounds right. So, delete "between you" and replace with "for." Would you say "This question is for I." or "This question is for me." Me is the answer!! "You and I should go to the park tomorrow." Delete "You and". "I should go to the park tomorrow" or "Me should go to the park tomorrow"? "I". Finally, "Yesterday was a really good day for her and me." Remove "her and". "Yesterday was a really good day for" "I" or "me"? "Me". This last sentence is a double issue as whenever we use she/her or he/him together with I/me, this issue appears with respect to those pronouns as well. Many people will say "Yesterday was a really good day for she and I.". In part because "She and I had a really good day yesterday." seems like the same sentence. It is not. In the sentence beginning "Yesterday", she and I are the objects of the sentence. In the sentence beginning "She", she and I are the subjects of the sentence. Again remove the pronoun and preposition, but this time remove "and I." "Yesterday was a really good day for she." Wrong, "her" is correct. Remove the "she and". "Yesterday was a really good day for I." Wrong "me" is correct. Again, sorry for the length, but this is a persistent issue.

"Between you and me" is correct because the rule is that pronouns following a preposition are objects of the preposition. In this case the preposition is "between" and therefore we use the objective pronoun "me" instead of the subjective pronoun "I". In a similar sense you wouldn't say "between we" but rather "between us" because "we" are not active and are therefore subjected to being the object (bad joke). (Sorry, I accidentally submitted this as a comment!)

Between you and me. Come on, Jiao, you know that after a preposition you use "me".

"Between you and me" is correct because the rule is that pronouns following a preposition are objects of the preposition. In this case the preposition is "between" and therefore we use the objective pronoun "me" instead of the subjective pronoun "I". In a similar sense you wouldn't say "between we" but rather "between us" because "we" are not active and are therefore subjected to being the object (bad joke).