Looking for train or bus service recommendation. Prefer not to rent a car.
Fly from Naples after you have taken the Cureri bus from Sorrento. Use one of the lo-cost inter EU airlines that you can find on Whichbudget.com or Skyscanner.com. Be cognizant of luggage restrictions.
Local train services between Sorrento and Naples are operated by Circumvesuviana Ferrovia. Trains leave Sorrento roughly every 30 minutes, the trip takes just over an hour and costs €4.10. See http://www.eavcampania.it/ Naples to Bari will be a Trenitalia service, so use their journey planner at http://www.trenitalia.com/ to see the options. There will be one change of train, minimum journey time around 3 hours 50 minutes and prices in the range of €30 to €40, less if you purchase well ahead of time. For departures tomorrow I can see that some of the faster trains are sold out, which suggests it is a good idea to book in advance.