For the previous commentators: Indeed, genetive case would not call for the use of the word "I," which is nominative, used as ONLY a subject (without exception) not as an object.
To answer your question SIMPLY:
I = active (subject)
I - it is the one that DOES something.
Any other reference wherein something BELONGS to someone (to me) or IS DONE TO someone (to me) is objective case requiring the use of "me."
Me - passive/inactive
Me - it is always the one that SOMETHING IS DONE TO
Sadly, especially in certain reality TV, the average American has realized the notion of NOT USING "Me" as a subject. i.e. "Me and my friend are going to the store." This is great, however, what they have quickly and simply concluded is that the use of "Me" is always to be avoided.
If you're ever confused about WHICH pronoun to use, "I" or "Me" when another person is also involved (with you), i.e.:
"It was given to my wife and to me."
"My friend and I went to the store."
REMOVE the "AND" and one of the persons. Just as you would never say, "It was given to I."... you would also never say, "Me went to the store.
Similary, you absolutely cannot use "I" in "My wife & I's dinner..." ("I" IS nominative/subjective case.) You can REMOVE one of the characters and test the sentence. But, in this particular case, you must REPHRASE the sentence, using "OF" in place of the "'s" (they are equivalent to each other indicating POSSESSION, hence, the possessive pronoun (of))
"Both my dinner and that of my wife were delicious. Both of our dinners were great." "My wife's dinner and mine were great." We cannot always, not in every instance, put together a set of words or phrases correctly in English without rephrasing them.
BUT, we can, indeed, easily learn when to use "my/mine" instead of "I's," which is 100% incorrect and an abomination within English useage.
You only use "I" when you are active, when YOU are the one who is DOING something. You cannot make "I" into a possessive pronoun that replaces "my/mine."
Lastly, check out PREPOSITIONS and don't be afraid of them. Although there are roughly 50-70, the most frequently used ones to note are: of, with, in, for, from, above, around, beyond, under, etc.) You need not memorize them... you will instantly know them in the future if you create sentences with them, using "I" and "Me" by themselves. Later, you can add another person to the sentence "I" or "me") - and you WILL KNOW which to use! ;-) ... both of you living happily ever after.