When should one use "You and I" versus "You and me"?

When writing the subject of a sentence, use You and I.  When writing the object of a verb, use You and me. 

For example:

You and I are going to the movies tonight.  (subject)

The movies were fun for you and me.  (object)

An easy way to tell is to take the "you" away and read the sentence only with "I" or "me".

"Me are going to the movies tonight" makes less sense than "I am going to the movies tonight".

Also, "the movies were fun for I" makes less sense than "the movies were fun for me"

Take "you and" out of the sentence and the choice between "I" and "me" becomes obvious.

To be more specific, the "you and I" are part of the subjective case. The "you and me" are part of the objective case. The same applies to other pronouns as well such as "she" vs. "her".

For example,

 subjective case: She loves to go to parties.

 objective case: I like to take her to parties.