We loved the Kunsthistorisches Museum, not least because of its wooden floors: easy on the tired feet.

Sorry about the garbage! The edit function had a meltdown.

The Kunsthistorisches (Art History) Museum is among my favorite museums in the world! But it is large, so if you don't want to be overwhelmed, consider deciding in advance which sections you want to visit. And even if you decide against it, consider stepping into it's main hall / stairway for the stunning Klimt decorations.

If you are a fan of Klimt, the Belvedere is well worth seeing.

The Roman Museum (Römermuseum) doesn't look like much on the website but it was my favorite of all. Located below street level in the center of town with the sounds of modern life overhead, the impression one gets of the city literally built on top of previous civilizations is vivid and as it's right there, small & doesn't take long I recommend it even you aren't sure.

https://www.wienmuseum.at/en/locatio...mermuseum.html

The Hundertwasser Museum.

Agree with kja – I wouldn’t miss the Kunsthistorisches museum, which is a cultural treasure trove (the museum itself is a piece of art!). We also focused on a few highlights (useful brochure provided there with the "major draws" e.g. Van Dyck’s Samson and Delilah and Raphael’s Madonna on the meadow, Kunstkammer etc.) given the extensive collections. As you can see here https://shop.khm.at/en/tickets/, the kombi-ticket is a compelling buy and includes admission to the fascinating Imperial Treasury (Kaiserliche Schatzkammer Wien (https://www.kaiserliche-schatzkammer.at/en/ which otherwise costs 12€ separately).

In no particular order, these are the smaller museums I have enjoyed:
Museum of Heating Culture (I love the ceramic chimneys)
Tram Museum (I spend a fair amount of time on public and find the old trams charming.)
Leopold (modern, but not weird-modern art)
Chimney Sweep Museum
Museum of Illusions
Bank Austria Kunst Forum

Slightly outside of Vienna, but accessible on public is Museum Gugging with its Art Brut.

The big ones can be overwhelming without a strategy. The €44 annual ticket for the Kunsthistorisches Museum (Art History), for example, gives you unlimited access to Kunsthistorisches Museum Vienna, Imperial Treasury Vienna, Imperial Carriage Museum Vienna, Theatremuseum, Weltmuseum Wien (Ethnography Museum, and boy, those Hapsburgs brought home quite a few "souvenirs" from their world travels), the Imperial Armoury and the Collection of Historic Musical Instruments.

I have a love/hate relationship with Albertina. Their annual ticket is expensive, and I find myself wanting to see one-too-few exhibits to justify the cost. The bonus with the single-entry ticket here is that you also get to see the staterooms (which have a collection of Albrecht Dürer works, including Der Feldhase.)

As far as the museum cafes/restaurants go they are all overpriced and just "Meh" to me. These are the places I drop into for lunch near to some of the museums:
KHM/Armoury/Welt Museum: Glacis Beisl or the little place in Museums Quartier.
Just below the Albertina is the Augustinerkeller and their quite good Backhendl.
Not too far from Imperial Treasury is Beim Hofmeister. Classic Austrian in the former cow sheds of Empress Maria Teresia (we actually have a Stammtisch there!).
In the passage of Palais Ferstel near Kunst Forum is the always lovely French bistro Beaulieu (make reservations)
Adjacent to the Carriage Museum is a restaurant serving acceptable Austrian cuisine without the tourist crowds at the Schloss Schönbrunn restaurant.

Hope this is helpful.

See that no one has mentioned the MAK - Vienna's V&A. It has - or at least had - a well-regarded cafe.