Just dug out my notebook from my most recent trip.
Find a good bakery and buy some of the sunflower-seed encrusted rolls that everyone seems to sell. They're just great. With some butter and some cheese, it's a great picnic.
Visit the Prater, the old amusement park with the Ferris wheel built in 1897, and stop at Luftwerk, at the back of or just outside the perimeter of the park. I was staying with Austrian friends, and they brought me here -- and they ordered the most amazing spare ribs -- great, spicy, pepper-rubbed slabs that were fragrant with the smoke from the charcoal fires over which they cook. These were served with a warm cabbage salad (German style, with bacon and vinegar) that was simply outstanding. A really memorable meal. My friends told me this place has the same owner as Schweitserhaus, but is quieter.
Pastry is good pretty much everywhere. I will say that, on two trips to Vienna, I have found Sacher torte to be something of a disappointment. It's very famous, I know, but Austria is filled from top to bottom with great pastry, and the Sacher torte, while a nice little chocolate cake, just didn't blow me away. (Best pastry shops, however, were in Budapest, if you have a few free days to slip away.)
Demel is fabulous -- very elegant, very old world. They are most famous for their hot chocolate, and it is sensational, though pricey. You definitely want to have at least one cup -- I'm pretty certain hot chocolate at Demel is on at least one of those "hundred things to do before you die" lists.
Central Café is another of the very famous spots with an imperial past. It's near the Hofburg, so you can see it the day (or days) you visit there. (In the Hofburg, there's an extra charge to see the treasury, but it's worth it.) Anyway, Central Café is elegant, has newspapers in a dozen languages, and has some of the best pastry in Vienna. I recommend the Imperialschnitte and coffee. (It can be helpful, if you buy a phrase book, to learn the different terms for coffee, because Vienna is definitely coffee country -- and great coffee.)
The Naschmarkt is a Viennese institution. It’s in a great location for sightseeing, but the market, which dates back to the 16th century, is a destination in its own right. It is a great place to shop for goodies, from candies to fresh fruits and veggies, and to have lunch at one of the many stalls, coffee shops and restaurants.
There are a lot of unusual mushrooms available, so look for mushroom-based side dishes and even main courses on menus (pilze in rahm is a variety of wonderful variety of mushroom in cream sauce that I had last trip –- but it was only one of maybe four or five varieties I had).
Which reminds me – even though I wasn’t terribly excited about Sacher torte, the Hotel Sacher has excellent mushroom soup – and the place is simply elegant, so worth visiting – and even if it isn’t sensational, having Sacher torte at the Hotel Sacher is cool.
Ham (schinken) is different from our ham, too, and is worth trying – less salt, a little sour, very good.
Noodles with cabbage are a classic dish, and particularly popular at the Heurigers, the places out in the vineyards that serve new wine.
Of course, weiner schnitzel, but possibly the best known dish in Vienna is Taflspitz. It was Franz Josef’s favorite dish. It’s basically just boiled beef, but it’s served with a number of sauces (I remember a spicy tartar sauce and apple puree with horseradish) that elevate it about the ordinary. And it’s just so Viennese.
Well, that should get you started, but still leave you some room for your own discoveries. Have a great trip.