You want my opinion. Tackle Italy by dividing and conquering like the Romans did.
North or South Italy.
Don't try to be North to South...well, I guess you could but focus on areas specifically in the north and then south.
Here is my North Italy divide and conquer tip.
Start in Venice... Stay there at least 3 to 4 nights. Get lost.
Then take the train towards Cinque Terre, 3 to 4 nights and enjoy! These are the pictures you see in travel channels.
Then take a train to Florence (Firenze) and use it as a base.
From there you will hit, San Gimignano, Pisa, Orviedo, Siena, Pretty much several awesome (lack of better word), must see cities. Pretty much day trip cities. Of course, don't forget about Florence. How many days, is up to you, you can do two cities a day or 1 city a day but Florence alone is at least worth 3 to 4 days (especially, if you are an art buff). This is where art came from.
From there, since you already have a rental, drive down to ROME. There is a main highway but take the highway halfway and take a parallel local road towards Rome the rest of the way. What do you get?
A scenic awesome drive through Tuscany, talk about scenic, simply wonderful. Stop at a local vineyard, have a half bottle of wine and just enjoy.
I should note, there are several "cottages/houses" I don't know the proper term, where you can stay at some of these places for a night or two or more and live a tuscan life. Cook breakfast in the morning with your kids, there is a whole package for these things (expensive perhaps) but it's very nice "family" thing to do. This is an option you can do if you don't just want to run through Tuscany.
Once in Rome, stay there at least 3-5 days, depending on how much history you want to absorb.
That is northern italy for you. Of course, there are tons more other places but like anything else, divide and conquor. try not to sweep Italy in one fell swoop. Although 36 days is a good amount of time, divide those days to specifically be either north or south and then based on the days, that should help you plan accordingly.