That area is likely to be very hot in mid-June, though July and August would be worse. Be sure your car and your lodgimgs are air conditioned. Read hotel reviews with an eye to comments about room comfort during the summer.
I like the baroque town of Lecce a lot. There are many picturesque smaller places. I recommend picking up a comprehensive guidebook covering Puglia and making a list of potential stops and asking for input on them. ViaMichelin is probably your best online source for driving times, but remember that they don't allow for stopping, getting lost, searching for parking, or walking from your parking place to the attractions you want to see.