Vienna and beyond

Two day trips to consider , Klosterneuburg Abbey , just north of the city , and Baden bei Wien , just south of the city , are good choices . If you have not visited Kirche am Steinhof ,on the grounds of a large hospital just in the northwest of town , it is a must. Only open on weekends, it is Otto Wagner's Art Nouveau masterpiece, and will knock your eyes out . The 48A bus from the Ring to Baumgartner Hohe , takes you there . This video is a must for a good overview of the city's history .https://youtu.be/AFZBOTgL_Hk

That video is instructive, but is nearly 10 years old. The city has changed a bit during this time.

This reminds me of an old joke about Vienna:
If you are afraid that Armaggedon is near, settle in Vienna, because everything happens there 10 years later.

Also , this book is required reading for anyone delving into the depths of Vienna https://en.m.wikipedia.org/wiki/Fin-de-si%C3%A8cle_Vienna

You mentioned Melk, but if you haven't been to the other Wachau valley towns (Dürnstein for one), they also make an easy day trip. Nice bike, bus or boat rides along the Danube are possible.
Same with the Wienerwald - as previously mentioned, you could visit Klosterneuburg, Stift Heiligenkreuz/Baden bei Wien, Mayerling, Laxenburg etc.
Other close-by places (<1.5 hrs): Eisenstadt/Schloss Esterházy (nice gardens), Rust on the Neusiedler See (lakeside town known for wine and storks), Carnuntum (Roman ruins), Sopron or Győr in Hungary.

Slightly further afield, Brno is accessible in 1.5-2 hours by train or Flixbus. More of a city than a "charming town" though.
Graz is an interesting and underrated city, although 2.5 hrs each way might be pushing it for just a day trip (although I know people who ride trains for 3 hours each way to visit the mountains here in Switzerland for the day, so never say never!)

What I like is the Neusiedler See area because very fascinating nature and good wine.
I think there is a train from Vienna to Neusiedl am See. From Neusiedl buses operating around the lake.
https://www.neusiedlersee.com/en

This article will give you an overview of Steinhof , and as I've written , this should not be overlooked . https://en.wikipedia.org/wiki/Kirche_am_Steinhof

Wow so many suggestions for things we’ve not seen in our previous visits.

Steven and wmt1: Thanks for the links to the Kirche by Otto Wagner. We have seen some of his architecture and other Jugendstil buildings but not this one. I will watch the linked “historical”, 10 year old video of Vienna. I requested the book through inter library loan.

Hopper 18: We did take the trip to Melk Abbey but the boat connections didn’t allow much time for the other towns so this time we will plan for those. Since we’re actually arriving in very late June maybe there will still be apricots on some of the trees in the valley. I know there will be plenty of apricot creations to eat.

We had so many heurigen within walking distance of our previous home exchange in Perchtoldsdorf that we never visited the larger, more distant northwest wineries. This time we will explore by tram and foot.

We were in Brno in 1998 just to see the monastery and pea garden where Gregor Mendel lived and worked. All of the little hand written signs were in German and we could walk around in the garden. I enjoyed seeing the monastery’s library. This time, we will revisit the monastery to see if its been updated for visitors and see the city of Brno itself.

Mignon: We will also go to the nature area south of Vienna for an outing and to see more storks, a passion of mine. I may even try to see the famous nesting grounds on an island east of Budapest on our multi night trip there.

Thank you all so much for the suggestions so far. We knew we needed to spend some more time in this beautiful and historically rich area and we are excited for this trip.

Mona , a few words about the Schorske book - these are seven essays about political and cultural life at the turn of the twentieth century. They do not need to be read in front to back order . The chapter pertaining to Gustav Klimt " Gustav Klimt : Painting and the Crisis of the Liberal Ego " is singularly the highlight of book . It tells the story of Klimt's clash with the conservative establishment over his commission for the University Ceiling Paintings, particularly that of " Medicine " . Equally important is " Politics in a New Key : An Austrian Trio " tracking reactionary anti semitism in Vienna and the position of Theodore Herzl during that period . Take your time with this book , it will enrich your experiences immeasurably.

Mona,

RE: Kirche am Steinhof:

Although your return to Vienna is still a number of months away, please check details at the Wien Museum's website in Englsh, particularly about entrance/approach to the church proper and when the church interior is open for guided tours.

When I visited Steinhof a few years ago, I approached from the south and walked uphill through the hospital grounds. Apparently, that is not possible at present, as the Wien Museum states the approach is from the north via a nearby fire station. The Klinik Penzing's hospital facility website states similarly (in German).

RE: animals.

I like turtles, bees, and ground rodents (re. European ground squirrels), all found inside a big park operated by the City of Vienna and free-of-charge to enter. I really liked the Blumengärten Hirschstetten n the 22nd district. I still can't believe there's no admission charge ...

Another big green space I liked that's also free to enter is in the 13th district for the Lainzer Tiergarten (Lainz Game Reserve) where deer and boars roam.

RE: architecture.

This was a massive theme I explored during a month in Vienna earlier this summer; I looked specifically at Otto Wagner, Josef Kornhäusel (Biedermeier period), and Theophil Hansen. It's not difficult to make your own tour for each of these architects. Because they'll come across your path, it becomes a mounting task then to include architectural works by Johann Bernhard Fischer von Erlach, Lukas von Hildebrandt, Eduard van der Nüll, August Siccard von Sicardsburg; and more "modern" works by Margarete Schütte-Lihotzky, Zaha Hadid, Hans Hollein, Adolf Loos, Fritz Wotruba, etc.

This book is already six years old, but it may give you some hints:
https://www.amazon.com/Places-Vienna-That-Shouldnt-Miss/dp/3954512068/

I spent a couple weeks in Vienna last year and made several day trips. I agree with the recommendations of day trips to Sopron, Hungary and Eisenstadt. I loved Graz; although it would be a long day trip (I spent two days there) it would be worth the effort as it has a beautiful and lively old town and the stunning Schloss Eggenberg, both Unesco sites. I would also recommend the easy day trips to Linz and Steyr, both in Upper Austria. Linz is a pretty city on the Danube, and I enjoyed the lovely Klosterhof beer garden in the center. Steyr is a picturesque, historic, hilly city where the Steyr and Epps rivers meet. It's an undiscovered treasure.

Henry and Bill: thanks for the additional, interesting locations and tips! We will check them out.

Wmt1: thanks for the book reference. We actually used a book like that for a different city this summer. Our hosts had it for their guests. While we didn’t have time nor interest to explore all 111 unique things about their city, we read all of it and used some for exploring things that don’t get into the standard guidebooks.