Museum of Applied Arts, Vienna

Design and fine crafts are the main features here, with exhibits of furniture, glass, china, silver, and textiles from the Middle Ages to the present day. 

Naschmarkt

Vienna's famous outdoor food market: Cheese, fish, olives, falafel, fruit & veg, etc etc, plus cafes and restaurants too. You can get a legit light lunch from all the free samples you'll be offered, but beware, they're wily businessmen around here! It's particularly worth being here on Saturday mornings, when there's a substantial antiques / fleamarket next to Kettenbruckengasse station. Neighbouring streets on the right side (heading towards Karlsplatz) also house a smattering of Japanese / Korean joints, if you've been craving ramen / sushi / kim-chi.

Kunsthistorisches Museum Wien

AKA 'Kunsthistorisches Museum'. Enormous (seriously, ENORMOUS) art collection amassed by the Habsburgs and housed in absurdly ornate surroundings. You could spend a week here and not get close to seeing everything.

Belvedere Palace

Another of Vienna's main sights - A collection of Austrian art (including much of Klimt's work), in the grandest of surroundings. The manicured gardens are free to walk through. 

Schönbrunn Palace

Former Summer Palace of the Habsburgs, Schönbrunn is on the UNESCO World Heritage list and is one of the most popular destinations in Vienna. Various tours are available of the Palace itself (priced from €13.30), but wandering around the gardens is free and well worthwhile, particularly the ten minute walk up the hill to the Gloriette, for great views across the city. 

Schönbrunn Zoo

The oldest zoo in the world. All the usual suspects, plus the main draw: Giant Pandas. And if you've seen those 'No Kangaroos in Austria' t-shirts in all the tourist shops, you'll find out it's a lie ;)

Museum Quarters Vienna

Hip cluster of museums; highlights including two of the big players in the modern / contemporary art scene, The Leopold & MUMOK, plus the Architecture Center and a few others. Depending on the time of year it can be a popular place for a beer in the evening, too, whilst lazing around on the iconic 'Enzi' chairs.

Hundertwasserhaus

Something a bit special: An apartment building designed by the fascinating Austrian artist Friedensreich Hundertwasser, with an emphasis on bold colours, the incorporation of nature, and a rejection of straight lines. There are strong similarities to some of the Gaudi buildings you might've seen in Barcelona. If you like this, check out the nearby 'Kunsthaus' (a few blocks over), also designed by Hundertwasser.

Kunst Haus Wien. Museum Hundertwasser

Another building designed by eccentric Austrian artist Friedensreich Hundertwasser (see also Hundertwasser House). This one has an equally outlandish appearance, and houses a museum on his life and work. Check out his wikipedia page - He was a cool guy. 

Hofburg Palace

One of the Must-Sees in Vienna - A Thirteenth Century former imperial palace, most famously the Winter residence of the Habsburgs. Wander around outside and admire the obscene grandeur for free, or do a tour of the Imperial Apartments, the Sisi Museum and the Silver Collection (Audio guide €12.90, human guide €15.90. Students get a whole Euro off). It's a beautiful place to wander through in the evening, too, when it's all lit up.

Albertina

Major gallery, the main collection being 'Monet to Picasso', an impressive selection of 19th/20th Century art. 

Amalienbad

When it was built in the 1920s, the Amalienbad was one of the largest public bathhouses in Europe - today it’s known for its lovely art deco architecture. It includes a large swimming hall as well as men’s, women’s, mixed, and ‘family’ saunas. 

Wiener Staatsoper

Wiener Staatsoper is one of the leading opera houses in the world. Its past is steeped in tradition. Its present is alive with richly varied performances and events. Each season, the schedule features 350 performances of more than 60 different operas and ballets. Every night, you can witness world-class artists alongside the permanent ensemble members on the stage and at the conductor’s desk, accompanied by a unique orchestra: the orchestra of Wiener Staatsoper, whose members at the same time make up the Vienna Philharmonic Orchestra.

Stephansdom

AKA Stephansdom, an enormous gothic cathedral in the heart of the old town. You can enter the main hall for free, pay €4.50 to climb the 343 steps of the south tower for views over downtown Vienna, or tour the catacombs (every 15-30 mins, €5.50).

Central Cemetery

Prater

An old skool funfair, an iconic (and disturbingly old!) ferris wheel, and an expansive park to stroll  (or better, cycle) through with the Viennese at weekends.