Thanks to Chowhound I had a very tasty trip to Vienna (special thanks to Sturmi). As a thank you I wanted to give you a recap so you can all pat yourselves on the back for your great recommendations!
Day 1. My first stop was Demel, which looks like a shop I would expect to find on Diagon Alley. I left with an apfelstrudel and sacher torte. Dinner at Zur Goldenen Kugel-- we don't speak a lick of german so the all-german, enormous, handwritten menu proved a challenge. I ended up with the goulash special and pumpkin soup (the best I had on the trip, same can be said of the potato salad) and my husband got the schweineschnitzel. This was my first ever goulash and it was delicious. Great, authentic way to start out the trip.
Day 2: Dinner at Gasthaus Poschel-- Our favorite! We ended up going there again on our last night in town too. Cash only, so be prepared. Very friendly waitstaff, obviously a local place. The food was fantastic. We split the tafelspitz the first night and then went for schnitzel and spaghetti on our last night. (reservations recommended)
Day 3: We started at the Museum Quarter and had lunch at Cafe Corbaci. I had some kind of creamy noodle casserole which was outstanding. After a trip to MUMOK we stumbled upon some huge festival near Hofburg. We got some pastries, pretzels, beer and wine and listened to a polka band play Beatles music. Dinner at Phonixhof-- Very pleased with this pick. There was no wait at 8:00, walked right in. Had great stuffed schnitzels (I went with ham feta and garlic, husband had cordon bleu).
Day 4: After a trip up to Kahlenberg we went out the Heuriger Wieninger. We got the four course tasting, which included a very nice venison (something I rarely get to eat in the US). Very tender, nice sauce, over some mushroom ravioli (I discovered that Austrians love their chanterelles). There was also a very good pumpkin soup.
Day 5: This was our Schonbrunn/Zoo day so we ate lunch at the beer garden in the zoo, which was pretty decent for zoo food. We then went up to Cafe Gloriette for coffee and cake. We split the apfelstrudel and Gloriette cake. Fantastic. We were super tired and ended up with Kebap near our hotel for dinner and then snacks from the hotel shop later.
Day 6: After a long sightseeing day we went to Foderboden for dinner, which I found on yelp rather than chowhound. Again with the all-german menu, but I made a cheat sheet after our first night so I was prepared. We split a chicken salad (all the schnitzel was getting to us). Very fun atmosphere (obviously some big party going on upstairs). Great outdoor seating. It is off the beaten path but not far from our hotel. I would recommend it for casual dining.
Day 7: FINALLY, we had reservations at Silvio Nickol's. Glorious. Absolutely the best meal I've ever had in my life (AND the most expensive). We did the 6 course tasting. We were bowled over by the appetizers. My favorite course was the lamb (pictured), which came with eggplant, mushrooms and dill (I don't even particularly like dill and this was great). The scallop in the first course was also divine. We got surprised by some canapes at the end in honor of our anniversary. It was a truly unforgettable experience. The only complaint I have is that the very young waitstaff was sniggering an awful lot and made us somewhat paranoid and uncomfortable. Could have been unrelated, but given the extremely high level of service we received, this behavior really stood out. Certainly wasn't enough to ruin our good time.
Overall a fantastic trip. Thanks to everyone who helped me out!