Austrian food is distinctive and delicious, and is traditionally of the stodgy, hearty "meat and dumplings" variety. Wiener Schnitzel (a bread-crumbed and fried veal escalope) is something of a national dish, and Knödel are a kind of dumpling which can be made either sweet or savory according to taste. In Vienna the Tafelspitz (boiled beef with potatoes and horseradish) is traditionally served on Sundays, and is normally accompanied by clear broth with dumplings and herbs. Apart from these, Austria is renowned for its pastries and desserts, the most well-known of which is probably the Apfelstrudel.
Bread is taken seriously in Austria. Almost every village has its own bakery, offering a large choice of freshly baked sweet and savoury rolls daily from 6am. Rye bread (Vollkornbrot, Bauernbrot) is the traditional staple food among peasants. If this is too heavy for you, try the common white bread roll (Semmel). Somewhat surprisingly, it is easier to find good bread outside of Vienna, where the baking industry hasn't yet come to be dominated by industrial scale chain shops.
Some Austrians have a habit of eating sweet flour-based dishes (Mehlspeise) for a main course once a week. Varieties include Kaiserschmarren, Marillenknoedel, and Germknoedel.
The best advice is to dive into the menu and give it a go - there are no nasty surprises!
Drink
Vienna is famous for its café culture, and there are coffee houses all over the city, many of which have outdoor terraces that are popular in the summer. Visit them for coffee (of course), hot chocolate and pastries. Most famous is Sacher-Torte. Most likely you will find the "top" coffee houses at the Ring, the Kärnter Strasse, Graben, and some other (maybe difficult to find) places in the first district (Innere Stadt). There are also very nice old coffee houses a little outside of the Ring at the districts within the Gürtel. Please do never just order "coffee" as you find dozens of different sorts from Kleiner Schwarzer (small black coffee) and Großer Brauner (big coffee with cream), Melange (coffee with milk) to specialties like Kapuziner (small black coffee and a drip of cream) and Kaisermelange (coffee, milk, egg yolk and cognac) on the menu and the Viennese Ober (the "senior" waiter) takes pride in this variety.
Austria has also some first class wines, mostly whites, slightly on the acid side. Wine can be drunk pure or mixed with mineral water, called "G'spritzter" or "Spritzer". The best place to do so is at the "Heurigen" in the suburban areas of Vienna. Originally the "Heurigen" was open only in summer, but more recently you can have your "Spritzer" throughout the year with a little self-served snack.
Austria has also a national soft drink called Almdudler. It is lemonade with herbs. Other typical Austrian soft drinks are Holler or Hollundersaft. It's a soft drink made of elderberry blossoms.
Like in most of central Europe, a large number of high quality lager beers are brewed in Austria. Every region and bigger city has its own brand of beer. Stiegl is easy to find and is usually considered to be excellent, especially on American palates.
Schnaps is a type of fruit brandy served in many parts of Austria, usually after a meal. The most popular flavours are pear, apricot, and raspberry, though dozens of other flavours are available. There are three quality tiers of Schnaps: distilled, infused, and flavoured. The distilled variety is the highest quality; several brands of Austrian fruit Schnaps rank among the best in the world, but are accordingly expensive: a half-Liter bottle can cost up to €100. "Real" Schnaps is made from real fruit (either distilled of infused). Beware of the cheap stuff sold in large bottles in supermarkets; this is often of the "flavoured" type - nothing more than pure ethanol mixed with artificial flavouring. If you want the real thing, go to a deli or upscale bar (if you're in a bigger city) or a Buschenschank (Farmhouse) (if you're in the countryside). However, be careful with Schnaps especially if you are not used to alcoholic drinks!