‘You and I’ vs. ‘You and me’
For many English speakers, even native ones, it feels natural to say “you and me” (or even “me and you”), no matter how we use it. For others among us, it’s then natural to correct the first speaker by saying it should be “you and I”, which is often true, but not necessarily; sometimes that correction is erroneous, and the initial expression was correct. While it’s proper for the first person pronoun to come last, one of its forms is not simply correct while the other is incorrect — it depends on context.
The difference between ‘I’ and ‘me’ is that the former is used as the subject of a sentence (or possibly as the predicate nominal) while the latter is used as an object. This shouldn’t be news to any of us; it’s the same as the distinction between ‘she’ and ‘her’, or ‘they’ and ‘them’. But while this is a fact that most of us know, we often forget it in the moment if the first-person pronoun follows ‘and’.
The solution is simple: pretend the pronoun isn’t coupled with another and consider its place within the sentence.
For example, “You and me are going to have a great time!” is incorrect because we wouldn’t say “Me is going to have a great time!” On the other hand, “She’s coming with you and me” IS correct because we would say, “She’s coming with me.” To suggest that someone change it to ““She’s coming with you and I” is incorrect because we wouldn’t say “She’s coming with I.”
Whatever form the pronoun takes on its own is the same form it should take when paired with another nominal in the same phrase.
‘Who’ vs. ‘Whom’
‘Whom’ has the opposite problem in that it is more often underused (while ‘you and me’ is probably overused). But the solution is the same. ‘Who’ is for subjects (or predicate nominals), while ‘whom’ is for objects. So again, consider this pronoun’s role in the sentence (its part of speech) to figure out which to use.
Whenever preceded by a preposition, it should definitely take the whom form: “to whom”, “with whom”, “for whom”, “from whom”, etc.
Whom should also be used after verbs, in most cases: “They saw whom?” “You met whom?”
Use whom for object questions for which the sought answer is a person, named animal, organization, etc.
When I teach subject vs. object pronouns at lower levels, I include ‘who’ and ‘whom’, including in tables I give them like this one:
While students won’t use whom all that much in those early stages, this helps to familiarize them with it so that it’ll seem more natural once they do need to start using.