Café Landtmann is a classic coffee house good for coffee, cake, apple strudel, but also for Schnitzel (and I would have a Frittatensuppe first as a starter).
Salon Plafond, which is the restaurant at the MAK (Museum of Applied Arts - good for exhibitions, design, architecture etc), also does a very good Schnitzel, and Gulasch. Their cakes are nothing special though - if you go there for lunch, I would go over the road to the Cafe Pruckel afterwards for tea and cake: another classic coffeehouse that had a refurb in the 50s.
The Museumsquartier is a popular place to eat/hang out on warm evenings: there are a few cafes/restaurants there, unfortunately our favourite one changed hands but in any case it's still nice to have a drink there on a warm evening, as is the Volksgarten Pavilion (60s vibe).
If you fancy Japanese/Sushi, then go to Mochi on Praterstraße - amazing food. They also have a small canteeny type place called OMK in the first district (Hoher Markt) which is great if you want a pit-stop while you're in town, or an early-ish supper (closes at 9.30pm).
Another good place for a snack is Tresniewski on Dorotheergasse in the first district. Open/finger eggy/pickly sandwiches and a Pfiff (tiny beer). Absolutely classically Viennese.
If you go to Schönbrunn Palace and want to stop for coffee, I would avoid the Cafe Gloriette at the top of the gardens (though it's ok) and go to the Cafe Dommayer which is close by and beloved by the locals.
My friend tells me that the Spanish riding school has rehearsals most mornings that are a lot cheaper than going to an actual performance - might be interesting? (I haven't been myself).
The Naschmarkt is an amazing market (food) and on Saturdays there's a flea market at the other end of it too.
If it's hot and you want to jump into an outdoor pool, take a cab to the Krapfenwaldbad which is high up on a hill with views over Vienna. Or the U-bahn to the Schönbrunnerbad which is in the grounds of the Schönbrunn Palace - more expensive but less crowded and lovely.
The 19th district / Grinzing (where the Krapfenwaldbad is) is also where the Heuriger are - taverny type places where you drink the 'young' wine. They can be really touristy but non-touristy lovely ones are Zawodsky (bit more of a trek but nice views) and Hengl-Haselbrunner (accessible via tram) - both have pretty gardens. Go for lunch or dinner, order half a litre of white wine, half of soda and mix your spritzer accordingly! Food is usually self-service, roast pork, fried chicken, sauerkraut, schnitzel etc.
Parks are smaller than in London or Paris, for example, but pretty. The Volksgarten has rose gardens: next to it is the Burggarten which is nice to sit in (also the Palmenhaus there has a nice cafe). The Prater is huge (last time I was there there was an escaped pony running down the middle of it) and lovely. If you go, get the tram to the Prater Hauptallee and avoid the funfair at the start of it (unless you want to go on the Third Man's big wheel): head right when you get off the tram (south-east, I guess) and it's all trees and grassy. For art, history, architecture, music etc - there's tons of information online. There are big museums, galleries, the opera house etc., all easy to find out about (though if you have any questions, do ask). Rachel Whitehead's piece is at Judenplatz, that's always worth seeing.
If there's a chance you might be wandering around peckish late at night, you could order a Käsekrainer - cheesy sausage - from a Wurstelstand. I've only done this a handful of times in my 25 years and generally I think one needs to be a bit well-oiled to consider it. But it is a typical Viennese experience. If you fancy it, go to either the one on Hoher Markt or Albertinaplatz.
As I write all of this, I realise I could go on and on! These are all some of our favourite things to do, at least. Let me know if you have any questions about anything else.