In this context, "grandfather and me" is incorrect.
An easy way to figure out whether to use "me" or "I" in a compound subject or object (Jim and I vs. Jim and me) is to simplify and restate the sentence as if it was just one person involved in the action.
For example: "Jim and I walk on the beach." is correct because "I walk on the beach" is correct. "Me walk on the beach" is grammatically incorrect, so "Jim and me walk on the beach" is also incorrect.
"That gift is from Jim and me" is correct because "That gift is from me" is correct. "That gift is from Jim and I" is as incorrect as "That gift is from I".
You can also do the same sort of thing with he/him, she/her, we/us, and they/them.
Hope this helps.