Bari and its environs -

Bari - walk around the old center "Bari vecchia", see the Basilica di San Nicola, the archeological museum, walk on the lungo mare. You can eat in the old center, for example the Al Pescatore restaurant, grab a quick bite at the nearby road Corso Vittorio Emanuele, or eat the restaurant La Bella Bari in the center if you're shopping around. Eat lots of focaccia, panzerotti, and stracciatella (the delicious cheese that's inside the burrata), as well as raw fish and shrimp, muscles, sea urchin, and vegetable dishes. 

Trani - see the cathedral, the Ognissanti church, and the Castello Svevo (you could stop for a coffee in Giovinazzo, a small town between Bari and Trani on your drive there). If you want to eat while you're there, some restaurants: Il Melograno, Le Lampare al Fortino, Casa Sgarra, Osteria Frangipane.

Castel del Monte - 13th century castle. People usually combine it with a visit to the Parco Nazionale dell’Alta Murgia and then drive down to Matera. If you’d like to stop on the way, then do so at Altamura where they have a nice church and well-known bread.

Matera - unmissable! Stay one night. 

Hotels: Palazzo del Duca, Caveoso Hotel

Polignano a mare - nice for strolls, gelato, often has ongoing cultural events. Famous beach that is often photographed by foreigners. You can choose to be based here while you drive around to other towns.

Hotels: San Michele Suite, Relais del Senatore, Covo dei Saraceni, San Lorenzo Boutique Hotel & Spa. 

You can eat at Da Tuccino. The famous restaurant in the cave, Grotta Palazzese, is very scenic but the food is terrible. 

Monopoli - very nice town close to Polignano that you can visit or, like Polignano, you can use it as a base from which to visit the area. Beautiful beaches close to Monopoli.  

Some people choose to be based in this area, and from here visit the Valle d'Itria. 

Valle d’Itria -

Ostuni - "la città bianca", the white city is very nice. Walk around the historic center, Gothic cathedral, etc. It's busy and touristy in the evenings if you go during the high season. Has many restaurants and bars for aperitivos. Buy fruits, clothes, house things, etc., in their Saturday morning market.

Cisternino - cute old town known for meat restaurants. Have "bombette" in one of the meat restaurants and coffee and torta di ricotta con amarene at Cremeria History Vignola, overlooking the valley.

Martina Franca - another nice town to walk around in the historic center, visit the churches, gates, etc.

Locorotondo - small charming town, great sunset views of the valley.

Alberobello - very touristy because it’s the most well known one for the trulli.

Savelletri - aesthetically the least pretty of these towns, but it's well located, convenient as many hotels are based around here, and the town isn't overly packed. Also good for a sunset aperitivo and dinner.

Hotels in Valle d’Itria (you could stay anywhere and drive around):
Hotel Relais la Sommità

Borgo Egnazia
Masseria Torre Maizza

Masseria Torre Coccaro

Masseria San Domenico

Borgo San Marco

Fatuaria
Masseria Moroseta
Masseria San Nicola
Masseria San Giovanni

Beach: Torre Guaceto natural reserve
Torre pozzelle, gorgognolo, lido bosco verde, etc.

If you like serviced beaches, these are some that are nearby:
Le palme
Coccaro
Lido bambù
Lido pettolecchia
Le tamerici
Macramè (more for aperitivo)

I usually eat at home or at friends' places when I go so I incredibly don't have that many amazing restaurant recommendations. You can eat in all masserias anyways, and here are some restaurants in Valle d'Itria:

Casale ferrovia (Carovigno)

Già sotto l'arco (Carovigno)

Taverna della gelosia (Ostuni)
Masseria il frantoio (Ostuni)

Osteria del tempo perso (Ostuni) 

Osteria Piazzetta Cattedrale (Ostuni)

Ristorante Porta Nova (Ostuni)
La peschiera (Savelletri)
Antica lama (Pezze di greco)
Pescheria due mari (Fasano)
Dentromare (Specchiola)

Ristorante Garibaldi (Martina Franca)

Osteria del coco pazzo (Martina Franca)

Salento -

You can then drive down to the Salento, this is further south and the heel of the Italian "boot". It has baroque towns and beautiful beaches, both on the Ionian and Adriatic side.

Lecce - unmissable city known as the Florence of the Italian south. Great sightseeing, even at night when as it’s all well lit. It's all about the churches, but you can also see squares, the castle, amphitheater, etc. 

Restaurants: Racale, Osteria degli Spirit, Bros', Duo, Trattoria Le Zie

Otranto - nice town to visit after a day at the beach, see the castle and cathedral. Great sunset views.

Castro

Santa Maria di Leuca 

Gallipoli (not in August, too packed with tourists) 

Galatina - not on the coast, to stop at on your drive back

Nardò - not on the coast

Hotels:

Dimora San Giuseppe (Lecce)

Patria Palace (Lecce)

Pollicastro Boutique Hotel (Lecce)

Masseria San Lorenzo

Hotel Palazzo Papaleo (Otranto)

Masseria Montelauro (Otranto)

Masseria Muzza (Otranto)

Palazzo Daniele (Gagliano del Capo)
Masseria Potenti (Manduria, province of Taranto)

There are many beautiful beaches in the Salento region, including: 

Lido punta della suina

Punta pizzo

Punta prosciutto

Porto miggiano

Torre dell'orso

Porto badisco

Porto selvaggio 

Grotta della poesia

Sant'Andrea

Torre lapillo 

Baia dei turchi 

Santa Maria al Bagno

Porto Cesareo