Learn Turkish: Garsonlar yorgun ama kaba değil. - The waiters are tired but not rude.

No, ama works exactly like "but" in English. You just put it right between the two words or phrases you want to contrast. Here, it sits directly between yorgun (tired) and kaba değil (not rude).

Why do we use değil for "not" instead of a suffix like -me or -ma?

In Turkish, the suffixes -me and -ma are only used to make verbs negative (actions like "don't go" or "doesn't eat"). Because kaba (rude) is an adjective, you have to use the separate word değil to say "not rude".

The subject Garsonlar (waiters) is plural. Why doesn't the sentence end with değiller?

It can, but it is optional! In Turkish grammar, if the subject is specifically mentioned and already has a plural suffix (-lar or -ler), you don't need to make the ending plural too. Saying değil instead of değiller sounds very natural here.

Where is the word for "are" in this sentence?

Turkish doesn't have a separate standalone word for "am", "is", or "are". Normally, you attach a personal suffix to the last word of the sentence. However, for "they are", it is very common to just leave the suffix off completely. The "are" is simply understood from the context!