Drop off car or keep it for Padova environs?

It depends on what you plan to visit. If you want to see the Palladian villas in Veneto, then you will need it, since they are not in the towns. If you plan to visit the cities only, then you can do without. You can drop it off in any of those towns, but certainly I wouldn’t want a car in Bologna, which is a big city with a large historical center closed to non residents’ cars.

How many days are we talking about? Padova has easily more than a day's worth of sightseeing; I spent 3 nights (2.5 days) there last September and did get to do all the walking around I had hoped for. Vicenza has somewhat less to see than Padova. I haven't been to Treviso or Bassano del Grappa. If you're tight on time for those destinations, you won't have much use for the car.

I haven't researched it seriously, but I think there's bus service to some of the Palladian villas along the Brenta Canal--though it may be too infrequent to be practical.

@Roberto da Firenze, I hadn't gotten far enough to learn about the Palladian Villas. Pretty impressive! Are they all open for visitors, like with some kind of pass, or just some of them? The Wiki entry wasn't clear.

@acraven: we are staying in Padova for 3 nights and hoped to be able to see Treviso, Vicenza, and/or Bassano di Grappas during a day trip. Hadn't learned about the Palladian Villas yet, and they look like they could take a whole day themselves! The main thing we want to see in Padova is the Scrovengi chapel/baptistry/museum. As time permits, we hope to see some of the following as well: Basilica of St. Anthony, Prato della Valle, Padua Botanical Garden, Palazzo della Ragione, Battistero del Duomo di Padova, Piazza delle Erbe. (Like you, we may never make it out of Padova!)

do you like the works of Canova?

There is an excellent museum of his works, plaster casts of most, nearby in the hills at Possagno, Museum Gipsoteca Antonio Canova. https://www.museocanova.it/

I'd think it best to arrive by car - that's how I did it....

@Nigel, I can honestly say I'm not familiar with his work. The pictures of the sculptures looking beautiful.

it is even more beautiful and inspiring in person in 3 dimensions

Another way to see some Palladian villas is to take a Brenta Canal boat tour. They can be full or half days. Google for details. You are taken into some of the villas. We had a very pleasant experience, with a very good guide. After doing the full day, I would say half day is sufficient, especially given your time constraints.

Be sure you don't sign up for one of the canal cruises during a low-water period. You could end up paying cruise prices for a bus tour.

Good point, @acraven. I found a pamphlet put out by the tourist office that lists the villas around Vicenza, so we're going to study that and see if we want to spend a day going from place to place. (Or, at least, those open for visitation.) If we still have the car, it's likely doable to see several of them. It's looking like we might not drop off the car until either before heading to Bologna or at the Bologna airport so that we're not driving in the city center.

The other thing we're grappling with is whether to keep our planned 3 nights in Padova and 4 nights in Bologna, or make it 4 and 3. There seems to be a lot to do in Bologna, too. (We have the Italian Days tour scheduled for our last full day in Bologna.)