So you have 13 nights, May 10-22. I would do 6 in Rome, 6 in Sorrento and 1 in Naples. How wise of you to not try to see the whole country and spend only 2 nights each here-and-there!
Rome as a base gives you the opportunity to see a lot in this great city, plus consider a daytrip to Tivoli (Villa d'Esteis fabulous) and Orvieto. Rick Steves' Rome Guide will tell you all you need to know.
Sorrento gives you a nice base to see the Amalfi Coast (day trip on bus) as well as Pompeii and Herculaneum. Allow some time to just enjoy Sorrento, or maybe day trip to an island such as Capri or Ischia.
Your last day, transfer to Naples and see the amazing Archeological museum there and take Rick Steves' Naples self-guided walk. You'll want the R.S. Italy guide to help you in Sorrento and Naples.
All of this is doable by public transportation, well-explained in the referenced guidebooks. Have a great trip!