Since folks know that I regularly visit Vienna, I get asked from time to time for ideas when others are making a short visit to the city. First of all, take a look at the Vienna basics page. Then, if you have more than a couple days, there are other things you can consider. But if you only have a day or two, at least consider these...
Haus Der Musik
Great fun, especially with kids. A museum that is very hands on, full of sound and exploration. Haus Der Musik.
Kunst Haus Wien
This museum showcases the work of Hundertwasser, a wonderful 20th century Austrian artist. The work spans sketches, paintings, posters, architecture, and much more. The whole building is a living expression of Hundertwasser's philosophy. It is unique and beautiful. Note, Modays are 50% off.
Getting there: take the Straßenbahn N toward "Prater Hauptallee" and get off at Radetzkyplatz. Then walk along the direction the train departs until you see signs on the street for "Kunst Haus Wien". Follow those signs to the left. It is a 5 to 10 minute walk.
Kahlenberg/Leopoldsberg and the Vienna woods
The Vienna woods are where the Viennese spend their weekend afternoon walks. The part of the woods I am most familiar with is the part near my Grandmother's house in Grinzing, so of course, I'm biased and want you to experience this little corner of the world. The best views of Vienna as a whole can be had from Kahlenberg and especially Leopoldsberg. Some of the most enjoyable walks in the woods can be had around Cobenzl. All of these are easy to get to by straßenbahn and bus.
A nice walk would be to take the bus to Kahlenberg, admire the view, then follow the posted maps to the path that goes to Leopoldsberg. There you will find an even better view and a nice small restaurant for a bit of refreshment. Then you can take the bus back down the mountain from there.
Getting there: take the Straßenbahn 38 all the way to the last stop in Grinzing (wave at my grandmother's street as you pass the second-to-last station!). Walk toward the back of the train as you exit the station and across the street. There you will find the bus stop for the Bus 38A. Most 38A busses go to Kahlenberg, but a few will go all the way to Leopoldsberg. A few will stop at Cobenzl.
We love the gelato in Vienna. Since our long-ago favorite Fratelli's in Schwedenplatz changed ownership and was renamed we have decided that Tuchlauben is the best in town.
Getting there: Walk northwest from Stephansdom on Brandsätte Strasse. Turn right at Tuchlauben Strasse. The eissalon will be on the left in less than a block.
Ride the Ring, walk the Fussgaengerzone
A cheap thrill is just riding the Straßenbhan around the whole Ring of downtown Vienna's old city (there is a special ring tram that does this, note the 1 and 2 no longer circle the whole Ring). If you feel like walking, get off at Schottentor (the other end of that 38 line to Grinzing) or at the opera and walk in toward the middle of the old city. You will eventually end up at Stephansdom, the old grand cathedral of Vienna. Much of this part of the old city is now a "fussgaengerzone" or pedestrian mall.
Jewish Museum of Vienna
One of the more hidden treasures of Vienna is its Jewish Museum. Vienna drove off most of its vibrant jewish culture during World War II, but it has a much longer and darker history of terrorizing jews. This museum is a sobering and beautiful reminder of this lost culture. The museum is in two locations, but I particularly love the branch in Judenplatz which exposes the ruins of an old synagog.
Getting there: you'll have to dig a bit more for this one. Check out this map (you may have to scroll around to find things).
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Sandwiches at Trzesniewski
Yummy tiny sandwiches for an euro each. Get at least three per person. We love the "ei mit ei" best, but there are dozens of varieties to choose from. Dorotheergasse 1 (take the 38 to the 1A and get off when the bus hits the fussgangerzone, then walk down toward the Stephansdom, keep an eye out for Dorotheergasse on the right).
Chocolate at Demel
Nathaniel's favorite on Kohlmarkt 14 (take the 38 to the 1A and get off when the bus hits the fussgangerzone, then walk up Kohlmarkt toward the palace).
Shop on Mariahilfestrasse
Take the 38 all the way in, then the 2 around the ring to the 2A, then the 2A up till all the stores appear. Then just get out and wander up and down the street.