Trip to Puglia May 1-May 15 Need advice!

Matera is only about 45 minutes (by car) from Bari, and there are bus and train connections, so it is possible as a day trip (a lot of your day trips are going to be that long by public transport). But, it is magical at night, so I would try to stay there. I loved staying at Torretta ai Sassi. It is very small--only three rooms--the owners are lovely, and the breakfast and view are both outstanding.
Bari and Lecce are the best bases for using public transport. If you are going to use scooter to get between places, I would not do it from either of those towns though, as getting in and out of them is pretty hairy. See if you can rent one in any of the Itria valley towns (like Alberobello)--riding among those villages over rolling hills will be much safer and much more pleasant. Near Lecce, you might want to rent one in either Otranto or Gallipoli for riding along the coast.

Why not just stay in Matera for one night? We much preferred Puglia to Matera but it was interesting to see it’s history although extremely sad, important to visit.

valadelphia, thank you so much for your reply. I guess I thought it was farther away. I do like the idea of staying to see it at night. I am especially thankful for your advice about avoiding the metropolitan areas of Lecce and Bari. I did choose those for the ease of public transportation, but was hoping to combine it with a scooter. I just love riding them on any given day but feel I would particularly enjoy it in Puglia.

I'm pretty comfortable riding in most areas as long as it isn't a large highway. May I ask what makes it particularly bad in those Metro areas? Is it crazy traffic or is it the necessity of using dangerous highways?

ETA - thank you for your reply too, Suki! Staying one night does sound more practical. And it will cut down on the nights of having to double book rooms as I already booked Bari and Lecce.

Your experience on a scooter means you'll probably be fine, but look at maps to get a sense of the routes you would take. Google maps is fairly accurate when it comes to travel times, adjust down for speed of course.
I have enjoyed riding scooters on islands--in remote areas where traffic was less of a concern, so that colors my response. When I am on one at high rates of speed for an hour, I find it stressful and kind of a drag. Rolling up and down bucolic farmlands or zooming along the coast is more like it! I would just want to minimize the unpleasant parts and maximize the good driving, so you could leave Bari on a train to Monopoli for example, get a scooter there and ride up to the Itria valley.
When I visited Bari and Lecce (two times), those two towns were definitely where I felt the traffic was more of an issue, and we were in a car. Basically everywhere else was quiet, rural roads--no worries there. I recall there being a route out of Lecce to the coast that was fairly quiet (it's the straight shot to San Cataldo), but I also remember a large oval traffic circle that we called "traffic anarchy."

valadelphia, thank you again! That is all really, really helpful. I don't mind driving fast for long periods, but I do want to avoid "traffic anarchy!" Your advice is already proving invaluable.

Of course the lazy rolling hills and beautiful coastline is preferred, but if I have to drive anyway I'd rather be on a scooter than in a car. Unless the weather is foul. In which case I will likely rely on public transport or stick to home base. I know I'm a bit different than most that way. But I'm also a bit different in a lot of ways, so I'm used to it. ;-)

Your trip sounds good to me. We will visit the area in June and also use Bari and Lecce as bases although we have more time and have booked 3 nights in Matera. You should definitely stay one night there. We are also traveling by train. Be aware that the main train lines like Bari to Lecce have many trains every day, but the regional trains don’t run on Sunday. There are no trains between Bari and Matera on Sunday although there is limited bus service.

This may be obvious, but make sure you book your tickets from Rome to Lecce in advance.
On our trip to Puglia, because of a missed plane, we ended up in Rome and tried to get a direct fast train to Lecce without success.
We had t get a slow train to Benevento, and pick up the fast train from there - but it did make for a very long, slow, journey.

It was explained to us that many people travelling from Rome to Naples catch the Rome-Puglia service even though the train doesn't actually go through Naples itself - they get off at Cancello and catch a local train into Naples. - but it does mean that the fast trains can get very crowded on the first part of the journey.

I will say I liked Puglia less than most other parts of Italy. That's not to say I didn't enjoy it - places like Trani, Gallipoli, Otranto, Monopoli and Policagno a Mare were all worth visiting. We didn't have a car, and at that stage some of the public transport was almost the definition of the word "Chaotic" - (particuarly on ferrovie del sud est (the local railway network)