Olomouc, Mikulov, or Znojmo

Mikulov and Znojmo are both lovely. Znojmo is more of a town, but Mikulov is more picturesque. They are both on the same (rather rickety) railway line. It runs from Znojmo to Mikulov and on to Breclav, where it connects to the mainline.

Jindrichuv Hradec is not a particularly good place to start if you're using public transport, as unfortunately, it happens to be in the middle of nowhere. A cursory glance at IDOS.cz (the public transport website) is listing Jindrichuv Hradec to Znojmo it as over three hours, with two connections. Given that its less than half that time by road, it might be worth your while negotiating a fare with a local taxi company.

Alternatively, it might be better to head back to Prague from JH and then take the mainline train to Olomouc.

I've visited all three of these cities/towns. I loved them for very different reasons. We visited Olomouc when we were using public transportation between Eastern CZ and Prague. It's a picturesque college town and the castle there was interesting. But, in my opinion, I really loved the southern region of Czech Republic and enjoyed the little villages, the wine country, and touring the castles in this area. I rented a car during this trip and it did make it really easy and our rental car was only about $24 per day.We started out in Brno and stayed two nights with friends and they took us all around Lednice, Mikulov (LOVED this little picturesque town), all around Palava and Klentnice - beautiful coffee shops (Cafe Ferra), churches, countryside and wineries. Then back to Brno and we toured St. Peter's Cathedral and ate at a little restaurant below the cathedral -L' Eau Vive. Walked around the town square, took in a farmer's market and then went and walked around Spilberk Park & Castle grounds.
Znojmo - stayed in Znojmo at a pension with a vineyard on the river (Kaplanka) - beautiful town with scenic views, incredible wines, and cute shops plus the Znojmo Castle and the Rotunda of St. Catherine was very, very interesting & beautiful with it's original frescos. Pro tip - don't park in public areas marked for residents. Got a ticket & boot on my rental car but the Czech police were very nice and the fine wasn't too hefty. We drove to Branisovice and ate at a cute restaurant (U Cisarske Cesty), and drove through picturesque Narodni Park, stopping in little towns and seeing the sites and churches.
So, it wasn't public transport, but I'd recommend Mikulov or Znojmo over Olomouc. Feel free to reach out with any questions!

Dana, thanks for your helpful reply! We will definitely aim for either Mikulov or Znojmo for our overnights and visit the other one. We will not try to do Olomouc. I will look into penzions in both places. We are planning to stay in Jindrichuv Hradec as that is the manor where my family is from...actually a village near there called Straz nad Nezarkou, but will be again visiting a lot of the villages in that general area. We are having great difficulty finding a car rental where we can get an automatic, as it would just be more convenient to get it in JH, but if we have to go to Brno, I guess we could. Navigating trains and buses isn't a problem since my cousin speaks Czech and we're adventurous, but we definitely want a car for some of our explorations to Znojmo and Mikulov. I will write back If I have any more questions.

Yes, I had a standard and I grew up without an automatic so it was a bit fun. Surprisingly, although I was apprehensive about driving, it was very manageable.

We rented at the Brno airport through rentalcars.com and got a really inexpensive $25 (equivalent) rate. I hope I'm not repeating myself but I would recommend not paying extra for the GPS and if your phone works with data, download mapy.cz and it was a terrific map. We used it while walking too and trying to find sites, parking, etc.

Happy to second Dana's shout for Mapy.cz. It's actually a better map platform than Google maps. Mapy is brilliant for cycle holidays too, as it has all of the cycle paths marked in.