Typical specialities are
- Wiener Schnitzel (the real one is from Veal not pork)
- Tafelspitz (cooked beef in a beefsoup)
- Kaiserschmarren (either as sweet main dish or as huge dessert)
- Apfelstrudel (Apple strudel in coffee houses or as dessert)
- Sachertorte (no doubt the most famous cake, in coffee houses with a 'Melange' coffee)
- Buchteln (speciality of Cafe Hawelka)
Wiener Schnitzel you will get most likely in every restaurant or pub (which is not specialised for other cuisine like Italian, Chinese, ...... food).
One place which is famous for Wiener Schnitzel is Figlmueller. You need to make a reservation at Figlmueller (one of their two restaurants) otherwise it will be difficult. The Schnitzel here is very big but at the same time very thin - so it looks more than it is but it is definitely enough for hungry people. The one in Wollzeile is the original, whereas Backerstrasse is a newer place around the corner but the food is the same. To do it right, you need to eat a Kartoffel-Vogerlsalat (Potato-Field salad) with the Schnitzel.
Figlmueller is a place for meat eaters only. There is nearly nothing for vegetarians.
If you want to experience the 'Tafelspitz' there is one very special (also a bit touristy) place to go. Restaurant Plachutta is specialised on Tafelspitz and the oldest restaurant is in Wollzeile (easiest to reach for you as in the city center). Tafelspitz is boiled beef and people really like it - not only Viennese people but guests usually really like it and want to go again if they visit Vienna the next time.
Again reservation will be necessary and in some cases you might even only get slots of 2 hours or so.
I am not sure how well served non-meat eaters will be.
Kaiserschmarrn, Apfelstrudel, .... there are no special places to recommend.
Kaiserschmarren (sweet dish with plum or appel compots- the dough is a bit like for pancakes) is something which you will get in many Viennese restaurants and pubs.
Viennese people might select it as a sweet main course too. If you eat it as a dessert, I would highly recommend to share it with somebody or ask for half a portion.
Apfelstrudel you will get in most coffee houses where you sit read newspaper or plan the next move to the next sightseeing place.....
Sachertorte - hmmm, you can get it everywhere and definitely in all coffee houses. For sure the famous 'Hotel Sacher' has the one which you can also buy in wooden boxes to bring home or send to people somewhere in the world.
Buchteln - I need to search the web to describe it...... and it is described as sweet dumplings with plum jam ..... I can understand the translation as the closest possible match, not sure I would call it dumplings. It is the speciality of Cafe Hawelka - which is a famous coffee house, where artists and students spend their time.
More hints and tips to come......