Best Base for 5 Days in Puglia

I would sort out the actual transport logistics first.
Like you, I prefer non-city locales when traveling, but I did not find Lecce particularly large. We like being about to walk to restaurants at night after exploring all day, so that's why I chose Lecce. Plus if the weather is bad I want to have stuff to do.
I only passed by Ostuni.
I loved Monopoli, which has great seafood and is well situated for exploring the coast and the Itria Valley. There is a nature reserve near Monopoli too, which had impressive dunes. Then there are lots of beach club type places with chairs/umbrella, etc--not my scene.
Did not make it into Gallipoli itself either, but it is closest to the nicest beaches we saw--there is a small nature reserve just north of Gallipoli, which was a highlight.

We like smaller places, too. And we weren't crazy about Lecce ... not that it was too big, but that it was practically a ghost town between 2 pm and 6 pm, and not a single place to eat until 8 pm. Not even pizza or gelato or a sandwich. A lot of Italian towns have the siesta tradition, but in most places, you can find SOMETHING to snack on. Not here. We arrived in Lecce, hungry, around 3 pm one day and were very annoyed by 8 pm, which probably colored our entire stay. Lecce does have some amazing baroque churches but we burned out on churches here too.

We loved Gallipoli. It is one of the most beautiful places we've seen. We just spent a day there and had a wonderful seafood lunch, with fish fresh off the boats.

But allow me to suggest an another option. In Puglia there are many masserias ... similar to an agriturismo in other parts of Italy. We stayed at one near Torre Canne, called Masseria de Montepoleone. It was centrally located, about a 15-minute drive to Alberobello, Locorotondo, Ostuni, Monopoli. So we could easily drive around in the daytime and visit places, then return and relax by the beautiful pool before dinner. They served amazing breakfasts, featuring their own products. They had a well-stocked communal kitchen, so you could purchase ingredients and cook there. Or drive a few minutes to a restaurant ... there was a great seaside restaurant in Torre Canne, 5 minutes away. It was a wonderful experience.

As for getting from Lipari to Puglia -- check out Rome2Rio for ideas. The simplest might be to ferry back to Milazzo, bus or train to Messina, ferry across to Villa San Giovanni and pick up a rental car. Then drive. By the way, we just spent 5 nights on Lipari in April and absolutely loved it.

I found the "pausa" to be in effect everywhere in Puglia--we almost starved on one scenic drive down the coast. Typically restaurants are the only thing open, but it is prudent to plan meals ahead!

Ostuni is possibly our favorite town in Puglia --- we just stayed there in March, in the old town, for a week. So many good places to eat. You will have to park your car outside the walls, but the airbnb host or the B&B host should be able to show you where.

We also loved staying at Masseria Aprile outside of Locorotondo --- you get to stay in your very own trullo next to the big house and the breakfasts were excellent. http://www.masseriaaprile.it/en/home/

I (without a car) loved Lecce. I prefer to stay in a larger place since I can't just hop in a car and teavel to the next town. With a car you can stay anywhere. I would consult at least one comprehensive guide book and make a list of towns to visit. Prioritize (musts vs. would-like-to) and mark on a map. Don't forget Matera in neighboring Basilicata. The map will help you identify the most practical geographic area(s) to consider. Use ViaMichelin to get a realistic idea of travel time from each projected base to at least your "must" destinations.

Many of the lovely places in Puglia are very small. Their points of interest can be covered in a few hours. If you choose a small town as your base, you will spend the majority of your daylight hours traveling to other places. You want that process to be reasonably efficient. I wouldn't want to be stuck traveling back and forth on the same road for 1 or 2 hours every day.

We also stayed for a week in an airbnb trullo outside of Cisternino. To me, it's sometimes a big advantage NOT to stay in a B&B where they give you breakfast ---- even though the breakfasts are great and the hosts are superbly hospitable and fun, we find that we are not be able to leave on a daytrip much earlier than 9:30am. In an apartment, we can leave at 7:00am and have coffee and crostata in whatever town we are going to first on that day.

In Ostuni, we had our son, daughter-in-law, 3-year-old grandson and 5-month-old granddaughter with us, so we had a strict rule not to go to any town that was farther than 45 minutes away. We drove to Polignano a Mare, Alberobello, Martina Franca, Grottaglie, Brindisi, and Ceglie Messapica. We broke the rule twice, to go to Lecce an hour away for the train museum, and to Noci to eat lunch when the places to eat in Alberobello looked pretty dismal. Except once, we always had dinner in Ostuni by just walking from our apartment for 5 to 10 minutes --- we find driving anywhere for dinner and then having to drive home kind of a pain. Without the children, we could have gotten to many other towns from Ostuni.

I just wanted to send a quick thank to everyone for your replies! I didn't receive a notification that people had posted so it was such a nice surprise to come back here, still having not made a decision :). I booked a Masseria outside of Martina Franca and forgot to cancel before it was too late to avoid fees so that helped make the decision for us! It is in an old trulli and has a pool, so hopefully we'll be central enough for at least a day trip to Gallipoli and all of the neighboring towns, but with room to relax if we want in the afternoons.

The help was much appreciated!