Italy: Palermo vs Amalfi Coast vs Puglia

Ciao a tutti. I'm travelling to Southern Italy with a few friends from early to late August. Spending a few days in Rome and a few in Naples, but for the next stop we can't decide on Palermo, Amalfi Coast, or Puglia. I've heard that Palermo is a great base and that it's a good place to experience local culture, nightlife as well as lots of pretty beaches surrounding it. I've heard similar things about Puglia except that the nightlife and "picturesque" aesthetic is a bit dampened. Is Amalfi worth it? It would be great to see because it's so close to Naples but I've heard so many negative comments about how expensive and crowded it is. Bear in mind: we are on a tight budget and in our mid 20s.