I still have problems to figure out when it's right to use I or me in an answer.

For example, what is the right answer to "Who did this?"

If I did it, should I say I or me? I've heard native speakers using both. Is there a rule?

And what are some examples when to use me, and when I?

UPDATE

Here's another example that I found today:

Took my Nexus 7 on a plane trip last week--first time ever. Downloaded movies (rental) from Google Play prior to departure. The size is much more convenient than even my old Netbook because it fit between my wife and I [...]."

Why isn't it "between my wife and me" in this case? Or is this just an example that the rules are getting fuzzy in colloquial english?