Learn Italian: Mi serve un caffè.

Yes, exactly! Just like mi piace literally means 'it is pleasing to me,' mi serve literally means 'it is useful to me' or 'it serves me.' The thing you need (un caffè) is actually the subject of the sentence.

If I need multiple things, does the verb serve change?

Yes. Because the things you need are doing the action, the verb must match them. If you need two coffees, you say Mi servono due caffè (literally: 'two coffees are useful to me'), just like you use mi piacciono for plural things.

Can I just say Io servo un caffè to mean 'I need a coffee'?

No, because io servo means 'I serve' (like a waiter bringing someone a drink). To say that you need something, you must use the backwards structure with the pronoun mi (to me).

How would I say 'he needs' or 'she needs' a coffee?

You just change the small pronoun at the beginning, exactly as you do with piacere. You would say Gli serve un caffè (He needs a coffee) or Le serve un caffè (She needs a coffee).