Is it "between you and me" or "between you and I"?

"Between you and me." After a preposition, English always takes the object form.

Is it "my friend and I" or "my friend and me"?

Depends on the slot. My friend and I went… for the subject; for my friend and me for the object.

Quick answer Canonicalizes to Me vs. I

Is it "me and him went" or "he and I went"?

i · AnswerOne line, no lecture

"He and I went." Subjects of a sentence take the subject forms he and I.

ii · A little moreWhy this is the one to keep

Doing the action of the verb puts the pronoun in the subject slot. You would never say "me went" or "him went," so you would not chain them either. Convention also puts the other person first — "he and I," not "I and he."

Related entriesKeep going

Less vs. Fewer For things you measure versus things you count. Usage
Who vs. Whom Subject versus object — the pronoun doing it versus the pronoun it happens to. Usage
Lay vs. Lie You lay something down. You lie down yourself. Usage
Under vs. Fewer than Position in space versus a count — ‘under’ isn’t always a substitute. Usage