You need a car, tbh, to reach the best beaches. You will also need swim shoes, and some of beaches have rocks.
DH is Italian; we have a house in Lecce but spend all summer at the beach near Santa Caterina. We go to the same lido north of Gallipoli each year, but also go to Santa Maria, Santa Caterina, Leuca and Sant’Isodoro.
We also spend a week at the Trullo in Santa Maria; no beach, but lovely swimming and diving off the rocks. V popular with teens, and they can walk into Santa Maria to the piazza for gelato etc. Santa Maria actually wouldn’t be a bad bet for you all; walking distance to both the sand beach and swimming off the rocks, plus a buzzing night scene in the piazza. There are some super aperitivo bars along that stretch of coastline, walking distance from the Quattro Colonne, that your children will love. There’s a bus service that services the lidos down to Gallipoli. For chicer evenings, Santa Caterina is posher.
The stretch of beaches from Gallipoli down to Leuca are stunning. Some are more suited to teenagers than others (we go to a family friendly one in Padula Bianca). Further down near Leuca, you have Torre Vado where you can do boat trips and have an aperitivo on Gregorio beach. Some lidos like Maldives are super luxe and very swish! Expect to pay between €25-40 per day for 2 sun beds and an ombrellone if you go to a lido. There are public areas, but tucked away and you’ll need the full kit of chairs, umbrella etc. It’s seriously hot and you cannot think of not having a sun umbrella.
I also love Porto Selvaggio; definitely go one day. Quite a walk down there, and you’ll need to bring your own packed lunch, but the water is pristine.
My view is that you need a car in order to fully explore the amazing beaches.