'Home Base' Needed

I think Verona would be perfect. Because you will have a car, consider Residenza Carducci. We've stayed there twice and can't praise it highly enough. Get the interior room with a kitchen. Here's my review on SlowTravel: ST review of RC

Here's their TripAdvisor page: RC on TripAdvisor.

Tell Giovanna that Michael from Seattle says Hi.

And if you want to see some of the reasons to visit Verona, check out some photos I took there in 2011 and 2013 here: Verona in May of 2011 and 2013

Verona is a great location, especially the old center of town. If you'd rather select a less-touristy city that may cost less for lodging, Padova is also a nice town with easy transport to both Venice & Verona. We've stayed at Moena in the Dolomites and really enjoyed the beautiful area.

You could pick more than one location to base from, staying a week or a bit longer in each place, and rent apartments instead of staying in a hotel room. Verona would be lovely. Another place you might enjoy for a week or so would be the Amalfi coast (plenty to see and do), or Sicily.

I would split my stay into more than just one location. Two weeks is a long time for staying in one place and wanting to visit many things. I consider it ideal to do day trips that are no more than 1.5 hrs drive one way (better if less than one hr each way). Anything more is really stressful on the driver. Driving in Italy is more stressful than in the US (even for me that I learned to drive in Italy). Roads are often very curvy and very narrow. City streets are congested. You must be continuously focused on the drive because you are navigating the car in very narrow spaces, often with just inches on each side. The minimal distraction will cause you to scrape the car against a wall or another car. Even on a freeway you can't fully relax due to the high number of trucks, the high speeds, and the fact that when you pass (rigorously using the left lane) you must make sure you clear the passing lane as soon as possible to give way to the faster vehicles behind you).

Roberto gives a nice accounting of some of the reasons that we avoid renting a car in Italy. Throw in the presence of speed cameras and the dreaded ZTL that can result in your receiving a ticket for 200 euros a year after your vacation.

I also think you should consider two locations if you'll be there for two weeks.

I think renting a car makes seeing different areas of Italy much easier than going by train or bus. We have never had a problem with Italian drivers and have driven just about everywhere in that country. I say go for living in Italy for your year. We live here for six months out of the year and are probably older than you and we are not rich. It has been a fantastic experience. I would say that if you are going to be there in the winter, to look into the southern parts of Italy or Sicily. Both places are very interesting and much warmer than any of the northern areas.

I would echo the Verona recommendation. It is central for what you want to see, has plenty to do while you are there, and is easy to get to and from. For Venice, don't rent a car - you'll just leave it parked the whole time you are there. Go via rail. For the Dolomites and the Lake region though, definitely rent a car. Don't let people put you off with their horror stories...we've rented plenty of cars in Italy and never once had a problem. I can't promise that you won't but not everyone has a bad experience. You can access the loveliest places in the Dolomites much more easily with a car - same goes for the Lakes. Just do your homework before you leave, understand Italian signposts and regulations about parking rules, and obey speed signs and other road signs. Even though you may find that drivers there tend to go quite fast, and will pass you frequently, just keep it slow and you should be fine.
We have done what you plan to do...stayed in one place for weeks at a time and explored the surrounding area. We love to travel this way and I think you'll like being able to just settle into a place, get to know a little better what life really is like there, and get to know a neighborhood. Great idea especially if you plan to spend an extended period of time when you retire. You'll figure out pretty quickly if it is REALLY what you want to do. Have a great time!

I also would split time between two home bases, to reduce that amount of backtracking along a similar path and to increase your range of logical daytrip destinations. Winter can be a good time to explore Italy with less crowds, and to attend Carnivale festivals, such as the battle of oranges at Ivrea, but not without rain or snow. Venetian ladies need to wear their fur coats, sometimes.