Me, Chris and Hannah went to the zoo. Now take out "Chris and Hannah" and you have Me went to the zoo. That does not work. It's supposed to be I went to the zoo. So the correct sentence is Chris, Hannah and I went to the zoo. Try this:
Mom gave pudding to Kate, Lisa and I. Time to take out the extra names, so the sentence reads Mom gave pudding to I. Nope. It should be Mom gave pudding to me. Then the other sentence is correctly said, Mom gave pudding to Kate, Lisa and me. (I find that people don't try to tell me I should say "I" when I put the "me" first - Mom gave pudding to me, Kate and Lisa.)
Please, please, please do not fall into the trap of thinking that if there's an extra name thrown into a sentence, it must follow with an "I".
Now get back to writing because I love reading!