Well, Puglia is my favourite region of Italy......
The trainline down the east coast which links Barletta, Trani, Bisceglie, Molfetta, Giovinazzo, Bari, Mola di Bari, Polignano a Mare, Monopoli and then Brindisi and Lecce is fast, modern and reliable and with a little patience public transport can get you everywhere except to the most remote beaches. This is why I love Puglia:
A blog about travel, vacations and holidays in Italy and Spain with advice, tips and trip reports
slowtravelitalyspain.blogspot.com
Here is an excellent trip report about seeing some of the most interesting places also inland using public transport
Happy to help if you need more advice or info....