Is It Me And Rick Or Rick And I?

It is Rick and I.

Actually, it depends on the context in which it is used.  One would say "Rick and I went to the movies." but "He was talking to Rick and me." is also correct.  There's a little trick you can use when you aren't sure if I or me is correct.  Say the sentence but leave out the other person.  In other words, instead of saying "Rick and I went to the movies", say to yourself "I went to the movies.",  or rather than "He was talking to Rickand me", you would think "He was talking to me.".  You wouldn't say "Me went to the movies.", and you wouldn't say "He was talking to I.".  I've been using this method for years, and it works.