“Jeff and me went shopping”

Vs

“Jeff and I went shopping”

If you can take Jeff out and it sounds right then it’s grammatically correct. For example you wouldn’t say “me went shopping”.

“That looks fake to Jeff and me”

“That looks fake to Jeff and I”

In that case you wouldn’t say “that looks fake to I”.

I never understood this until a technical writer I worked with made it so plain one day.

Edit: formatting

Yeah but unless it’s directly after a preposition you sound like a stuffy asshole. “Whom did I give the ball to?” Whom is falling out of usage in general and I won’t be sad to see it go.

And in some cases it’s difficult to line up the he/him, as in “Give this to whoever needs it.” In that case the whoever is almost pulling double duty of being the object of the preposition while needing to function as the subject in the clause “[subject] needs it”. But if you see the entire clause as the object of the preposition it works out with “whoever”.