Which is correct usage : "My sister and I" or "My sister and me" ?

It is not correct to say, for example, ‘My sister and me’.  The correct usage would be ‘My sister and I’.  

Example, “My sister and I went for a movie” is correct. “My sister and me went for a movie” is  incorrect.

However it is correct to use ‘and me’  in sentences where the phrase is an object, for example, “The president gave the award to my wife and me”. 

On the other hand, it is incorrect to say ‘between you and I’. The correct usage is ‘between you and me’.  The proposition such as between should always be followed by me, him, her, us ( objective pronouns).  For example, “Let us keep this secret between you and me”.