All you have to do is listen to each of the pronouns separately...
Give it to him...
Give it to me...
THEREFORE: Give it to him and me.
He went to the park...
I went to the park...
THEREFORE: He and I went to the park.
(AGAIN, the rule is that "I" and "HE" are subjective pronouns (subjects in the sentence); and "me" and "him" are both objective pronouns (objects in the sentence).