Should I say 'between you and I' or 'between you and me'?

The correct form is 'between you and me'. The word 'between' is a preposition, and prepositions are always followed by object pronouns, not subject pronouns. The object pronoun of 'I' is 'me'. So just as you would say 'between us' and not 'between we', you must say 'between you and me'. The phrase 'between you and I' is a common mistake people make because they think 'I' sounds more correct or formal, but it is grammatically wrong. A simple test: drop 'you and' and see what remains. You would say 'between me', never 'between I'. The same rule applies to 'for you and me', 'to him and me', and similar phrases after a preposition.