I live in Vienna and have a two year old, so I hope I can provide some helpful tips. First, I hear many people/friends complain that Vienna is not friendly to children. Personally, I have never experienced this, but I have to admit I am very careful about where I take my children when eating out. Restaurants in general are not very small child friendly here - seldom do you find high chairs or children's menus, for example. The following discussion might be a good start for you:
My strongest suggestion would be that you visit a heuriger. Heurigen are restaurants operated by small wineries which serve good Austrian food cafeteria-style. These wine taverns on the outskirts of Vienna are usually very child friendly - many have playgrounds making it possible to drink wine and eat while watching your child play. I am fond of Feuerwehr Wagner (http://www.feuerwehrwagner.at/), Bamkraxler (more of a beer place but has a kids brunch, etc; www.bamkraxler.at), Scheubel-Auer (no playground, but lots of space to run around; best food; www.scheubel-auer.at). If you want a really nice place, take the U4 to Heiligenstadt and then taxi a taxi to Buschenschank Nierscher, a heuriger in the Vienna woods with great food, mini-zoo and playgrounds (www.nierscher.at).
For a really nice Heurigen experience, you can also take the U4 to Heiligenstadt and then get on the bus 38A to Kahlenberg. This is a large hill/small mountain overlooking Vienna. Take the very stroller friendly walking path down the hill to Nussdorf (look for the signs with Stadtwanderweg 1, this will guide you down). Along the way you will pass some hillside Heurigen and get a great view of Vienna and the Danube. Sirbu is a particularly lovely heuriger (www.sirbu.at) along the way.
Back in Vienna, I would also recommend the restaurants in the Altes AKH, particularly the Universitätsbräuhaus where there is a playground attached to the outdoor eating area (www.unibrau.at). Cafe Einstein, just next to the Rathaus is quite child-friendly (www.einstein.at). Il Sestante has the best pizza in Vienna and a large outdoor space (www.sestante.at). If you go to the zoo, which you should, I recommend Landtmann's Parkcafe, at the rear zoo entrance next to the Neptune fountains (http://www.landtmann-parkcafe.at/en/parkcafe/).
Also nice is the Dschungel Wien restaurant and cafe located in the MuseumsQuartier right next to the Children's Museum. In a courtyard, so lots of space to roam (http://www.dschungel-deli.at/) and the menu is very child friendly. Strollers at every table.
The iconic Viennese cafes (Central, Demel, Sacher, Hawelka, etc) can be very difficult with small kids. Personally, I wouldn't go in there with my two year old - close quarters, generally unfriendly waitstaff (by tradition), too much to break, etc.