There’s a ‘standard’ grammatical error, whereby when a person is talking about doing something with another person they say “My brother and me went to the shops” or similar. To avoid this, generations of teachers have stressed the importance of saying “my brother and I went…” instead.
That’s not my peeve. The thing that annoys me is that various people seem to believe that the “my brother and I” construction should be used for both subjects and objects – so, for instance “they went to the shops with my brother and I.” The proper approach, of course, is to consider what form of pronoun you would use if it was just the first person singular: “I went to the shops”; “They went to the shops with me.”
Second peeve: so, this isn’t actually an error, but in certain circumstances it’s permissable to use either ‘who’ or ‘that’ to refer to people – my personal preference, in general, is to use ‘who’ (as ‘that’ seems to treat individuals as non-human objects), but I increasingly find people using ‘that’ when given the choice. Of course, I’ve also seen people use ‘which’ where they should have used ‘who’, but that’s an actual error, and not too widespread.