Hello timatspc, and welcome to the forum!
Either Verona or Brescia would work for what you describe. I have spent much more time in Verona but I liked Brescia when I visited there. Brescia buses serve the western side of Lake Garda while Verona buses serve the eastern side. I think Verona may have better fast train service than Brescia but as I said I'm more familiar with Verona.
Verona is a very pretty city nestled in the bend in the river Adige with an arty and cafe culture vibe. It is full of medieval buildings with faded murals and the 'elegant decay' that the Veneto does so well.
Brescia models itself more on Bergamo and has a series of connected but different grand piazzas, really under rated Roman ruins and a castle on top of the hill to top everything off.
Just to be clear when you say "visit" do mean a day trip or going and spending a few nights in these further locations like Turin? It's about 3 hours to Turin (or more) one way which well exceeds my threshold for day tripping. If you're thinking of day tripping I agree with the other posters that you're wishing for too much. There is no place close enough to all these places to effectively do this.
If you mean "go there for 2-4 days" and you're okay paying rent on both your 2 month location and your short stay location then I think you might be onto a workable plan. Otherwise it might be time to consider multiple bases or just longer connected stays in different cities.
Having just stayed in one place for a month one of my concerns would be getting into a situation where you're renting something sight unseen and it turns out not meet your expectations. AirBnB has erased the longer vacation market for real estate agents in Italy so you probably won't find anyone willing to work with you for such a short term rental. So you're likely stuck with AirBnB, VRBO or Booking.com as your only real choices. I reached out to a realtor based on an article he wrote on Italian rental terms and issues and he referred me to the rentals specialist who just sent me AirBnB links. All the sites I found that advertised longer term rentals received no reply or "We don't do that anymore but if you want to rent for 1 year+ or buy let us know." I loved my place but 2 months is long time to live somewhere that doesn't work for you.
Also be aware of seasons and festivals because they can drive up prices for specific weeks and weekends and that may limit your ability to rent if your two months covers that time. Verona has opera in July and August and various other events that can make summer rentals expensive and/or difficult. The very first time I visited there was some Italian singer I had never heard of doing three nights in the Roman arena and rentals that weekend were very hard to find and expensive. Brescia will probably be much less expensive since it is much less touristed but that may also limit your options.
I would also recommend learning some Italian. It is not required to generally get around but sometimes it really helps when you really need something. It also really helps if you are trying to live somewhere and experience being part of the community. Italians love to talk more than anything and if you can't talk with at all it really leaves you outside the main social function.
Hope that helps,
=Tod