|In the time you have you could hit the Barolo areas of the Piedmont (Asti, Alba), then to the Tuscany wines of your choosing, perhaps Sassicaia, chianti, Brunello, or Nobile, then down to Umbria for sagrantino or classic whites. It would be a great trip.
Let's see. Tuesday, Wednesday, Thurs, Fri, Sat, Sun.
Winery visits in Piemonte (BTW: Asti is not in the Barolo area... Alba is just next to it), tend to be multi-hour affairs, as they are not set up for Sonoma/Napa style taste and dashes. They will tour you the vineyards, the barrel cellar, the fermentation tanks, the whole 9 yards. They will taste you what they make. It may be 10 wines, as I experienced at Ada Nada in Treiso, where we stayed. It will likely come with some cheese and sausage, and be a whole morning or afternoon. Alternatively, there are regional enoteca (5, though Grinzane Cavour, Barolo and Barbaresco are probably the only ones worth visiting, as Mango is only open on F-Su and the Roero one is kind of out of the way. These can be banged out a lot more like a Sonoma/Napa tasting room. Additionally, some of the towns, like Nieve and La Morra, have tasting rooms connected to wineries for that faster style of tasting.
Tuscany, I have not done. But I'd recommend picking one region, not splitting time. There's a lot of driving in Piemonte (one of the most beautiful places I've ever been) and I'm sure it's similar in Tuscany/Umbria. It's a few hours drive from the one to the other, of not necessarily the most interesting driving.
I'd pick Tuscany over Burgundy, even though I'm more familiar with Burgundian wines. Tuscan wines, beyond Chianti, kind of demand a rosetta stone, for the sheer variety of grapes grown. Hundreds of varietals... I feel that Tuscany will produce wine of better value than Burgundy, but have no evidence to back that up.