Franconia Wine Tastings and Cellar Tours

What you should do is attend the Weindorf Wine Festival in Wurzburg which runs from 25 May - 4 June. Also, since it's Maibock time (in May) you may want to look around for beerfests. Ask your hotel staff, they can probably point you in the right direction.

The only winery in Würzburg that I know to offer wine tasting to individuals is the Juliusspital, the city's most prestigious winery. The May date is unfortunately already booked. But if you reserve a table in the restaurant and tell them how many wines you want to taste (and in which price range), they will surely arrange something for you and also provide explanations. The winery is in the middle of the city on the Juliuspromenade. I'm sure in the Bürgerspital (not far away) they would do the same, but why not try out the best.

@KGC
The 2019 Weindorf is from may 29 to june 16 ;)

This is a great area with a lot to see. Traveling by train/bus you will have to navigate schedules. Most small Weingut will give you a tasting regardless of the number of people onsite, but you should buy some wine too! I see some now do specify minimum numbers on their websites, but I haven't had a problem with two for tasting. You can always call ahead. Some of my favorite towns are Prichsenstadt, Volkach and Iphofen and there are comfortable places to stay in all. Check them out on the web. I buy wine in Prichsenstadt at Winzerhof Keßler. A great place for a Main river and vineyards view is Vogelsburg. Schloss Hallburg has a nice Biergarten. In May you will be there for Spargel (white asparagus) time too. Spargel with butter/hollandaise sauce, ham and boiled potatoes with Franken Wein outside on a sunny Spring day, it doesn't get much better.
https://www.fraenkischer-weinfestkalender.de/

So we will be in Franconia the week before the wine festival. I hope the Spargel is ready - sounds delicious! We are leaning towards lodging in Sommerhausen for the two nights.

You shall be on time for aparagus: For such dishes in Würzburg try restaurants Backöfele, Bürgerspital Weinstuben and Schützenhof (order A-Z). Obviously many more restaurants will have it one menu.

Sommerhausen is a lovely litte spot with cobbled backstreets, Iphofen a special spot. We were in the area the year before last. Part of Unity Day we were in Sommerach, which was very busy with locals. I'm not sure if this wine store is the local cooperative or not, but it sells many wines from the area and has inside/outside seating for tasting. A few places to eat at as well:

https://www.winzer-sommerach.de/Startseite.html

We are looking forward to hikes around Sommerhausen, visiting Wuerzburg on market day and stopping in Iphofen on our way home. Now to narrow down the wineries and restaurants to visit!