Baku is nice place for budget travelers. Tasty foods, combination of eastern and western architecture, cheap pubs! You can visit history, national literature and art museums which are very close to each other. I would say nightlife is not best.
Baku is not city which I would recommend. It is big and modern and clean, but it is visible that most of their sights were build couple years ago and therefore this city looks fake to me.
Night life and friendly people.
Delicious water - melon