Is 'Jack and I' or 'Jack and me' grammatically correct?

Which one is correct depends on where the phrase is found in the sentence -- specifically, whether it is acting as the subject or the object.

"Jack and I climbed up the hill" is correct because "Jack and I" are doing the action, so they are the subject of the sentence.

"A bear chased Jack and me" is correct because "Jack and me" is the object, since something is being done TO them -- the bear (the subject) is chasing them!

The correct grammar is Jack and Me and not Jack and I.