Close but not quite: the correct grammar is to always mention yourself last (with second person coming first if present in the sentence and then third person) and, as you say, use I or me depending on which one it would be if the others weren’t in the sentence.
Wrong
“Me and Dave are going spelunking”
“Remember that Dave, you, and I are going spelunking”
“This is a picture of Dave and I spelunking”
Right
“Dave and I are going spelunking”
“Remember that you, Dave, and I are going spelunking”