I want to explore Puglia, Basicilicata

For your type of trip I would recommend a car.
Since it is a lot of real estate you are covering, with lots of stuff to see, 4 weeks would be the minimum I would consider.
Sicily alone will easily necessitate 2 weeks, if you want to see most of it. Puglia and Basilicata will need over a week just for the major highlights.

Trains serve some destinations, but I would drive if your focus is on smaller towns. Service along the "sole" of the boot is by slower regional trains and buses, e.g. 12 hours from Bari to Messina, and the same on Sicily. (Simpler connections travel via Salerno but are not any faster.) If you need to make that long hop without other stops planned, it would be a good leg to fly. See several train company links on our Italy page, including on the "fine print" sidebar.

Is timing wide open or do you have a certain number of weeks? I would also say 4 or more.

You can also get some destination and timing ideas from Rick's guided tours but do allow more time if you're driving yourself, using slow train connections, or want to add more stops.

I would also consider just using a car for the "Boot" then switching to bus once in Sicily if the plan is the main towns.

I agree with 4 weeks as about the minimum. That's about how much time I had in that area, traveling by train and (mostly) bus in 2015. I spent about 18 days in Sicily, which was a bit short, and then didn't have enough time to see all I would have liked in Puglia. The only thing I saw in Basilicata was Matera, and that was a brief visit via one of the very handy van-trips organized by the the Lecce tourist office. I made no stops in Calabria at all. Having a car would bring efficiency and give you a good look at the area you've targeted. If you're seriously short of that, I'd recommend doing just Sicily or just the mainland.