Originally Posted by csschultz

I believe these are called subject and object pronouns. You would use I at the beginning of a sentence, such as I am going to the store. You use me at the end of a sentence, such as: Give it to me.

So, to answer your question, you would say, this is Joe and me.

Talking about your trip, you would say, Joe and I went to Hawaii. (beginning of the sentence) You might then say, This is a picture of Joe and me at the beach.

(If you break it up, it might sound clearer to you, This is a picture of Joe at the beach. This is a picture of me at the beach. You would not say "This is a picture of I at the beach, so you would not say, "This is a picture of Joe and I at the beach."

I hope that helps