Trastevere is the most atmospheric neighborhood — cobbled medieval streets, excellent restaurants, and a local feel despite the tourists. Testaccio is where Romans eat — less touristy and better value. Prati (near the Vatican) is practical for those focusing on the Vatican and has good mid-range hotels. Centro Storico (around the Pantheon and Campo de' Fiori) puts you at the heart of ancient Rome at a premium.
Avoid the Termini area for mid-range stays if possible (functional but soulless) unless you're a budget traveler — guesthouses there are cheapest. Boutique hotels in converted palazzi or convents offer unique Roman accommodation experiences. Book everything well in advance for April–May and September–October.