Wine and Dine in Montepulciano, Pienza and Montalcino

Plan to do your wine tasting while IN Montepulciano unless you have a designated driver or go with a tour
The alcohol limits in Italy are much lower than in US

Almost all wineries will require reservations
There are several shops right in town that offer tasting

If you love Brunello, then the Enoteca la Fortezza in Montalcino should not be missed. It has a huge selection of Brunellos to taste and buy. For individual wineries, I like Fattoria dei Barbi and Casanova dei Neri, but there are a lot to choose from.
In Pienza, we enjoyed Chiostro di Pienza's outdoor patio for dining, and nearby, in Monticchiello, is Osteria La Porta, with excellent food.

I meant to add, that in Montepulciano, Avignonesi winery offers tours, tastings, and a gourmet lunch paired with 7 of their wines, all overlooking the Val D'Orcia.
Not cheap but definitely worthwhile.

We liked Patrizia Cencioni winery/vineyard outside Montalcino very much - run by a woman and her daughters.
https://patriziacencioni.com
Casanova de Neri seemed more corporate, it's a lot bigger estate. I didn't like the tasting experience as much, though the wine was excellent!

We hired Fabrizio from Ciao Faby Tours to pick us up form our accommodations near Montalcino for a day of winery visits, tastings, and lunch at a sheep farm outside Pienza. Fabrizio (Fabry) was fantastic - warm, funny, and very knowledgeable, and definitely had a lot of local connections. I believe he also does cooking classes at his home in Montepulciano!
https://www.ciaofabrytours.com/best-private-tours-in-tuscany/exclusive-private-wine-tour-in-tuscany/

Our favorite meals in Montalcino were:
Re di Macchia
Il Grifo
Taverna Grapolo Blu

In Montalcino we ate at Il Moro for lunch and really enjoyed it. Not fancy, but good food and service. It's family run, and seemed like there were many locals eating there. We didn't have reservations, not sure they even take them, but had no trouble getting a table for 4 at about 12:30pm on a Tuesday in mid March.

Thank you everyone for your recommendations. I have a slight potential problem.

We are going to be in Tuscany from Friday night until Sunday afternoon. When I looked hours up a few months ago and clicked through the hours on Google, it appeared that most were open on weekends. However, now that I am looking at the actual websites, it appears that may not be the case.

Are wineries typically open on Saturdays in late May? I am slightly panicking because I don't have any wiggle room in our itinery and I am worried everything is closed while we are there.

We were in the area in 2022 and did a guided tour/tasting on a Sunday in July. No issues with going to the wineries that way. I'd guess if they were open for tastings on Sunday, more than likely they would on Saturday. My understanding is most wineries aren't "open" like in the US where they have their tasting rooms open set hours every day. Rather you have to book in advance with the winery.

Are you looking for a real wine tasting experience or just the best wines?
If you want a memorable experience, I highly recommend NostraVita Winery near Montalcino. It's a small family run winery. We were there almost 10 years ago and have done MANY wine tastings since but nothing compares to that experience. The owner (if he's still around) is a true Renaissance Man. He individually hand paints labels for each bottle and showed us how he did it. He also showed us his personally made collections of "books" which are actually containers of samples (typed story and oils) of all the trees in the area - each bound with the bark of the tree.
It is truly a family affair! He has 3 small vineyards - each named after one of his three daughters.
At the conclusion of our visit he led us in a toast to "Nostra Vita -- Our Life".
I would go out of my way to go there again!

If you want the winery experience, I don't wish to deter you. However, given the short length of your stay, you may find it better to do what we did, and try lots of local wines at a wine bar rather than wineries. It can give you a lot more variety.

In Montepulciano town, there is a great wine bar at the top of the hill, E Lucevan le Stelle, that has a lot of excellent regional wines by the glass and friendly English speaking staff. We tried several glasses there, all excellent. We also tried a wine bar a few blocks down, Opificio 16, but did not really like it -- the person working there didn't even know what wine she had opened and was very diffident. The restaurant Il Teatro (right off the main plaza) was another wine highlight of our trip, and we tried several different kinds of Vino Nobile by the glass, with fantastic guidance by the exuberant and very knowledgeable owner (and the best meal of our trip).

I also recommend the advice to hire a wine guide/driver. It really does make the day much more enjoyable and safer.

As others have mentioned, the best options for actually tasting wines are the in-town enotecas. I get bored with a winery tour already having done countless dozens around the world. However, a great enoteca may often offer different flights of the same varietals that really let you explore how several different wineries may have completely different taste profiles based on their terroir or their production methods. Much more informative.