People often ask me for recommendations when visiting or living in Zurich.
Here is a short list.
Travel: I tried to leave Switzerland at least once a month while I was there, and I'm sure you'll take advantage of the travel opportunities, as well. Two of the most underrated cities in Europe that I've been to are Ljubljana and Vilnius. I've had great holidays (with direct flights) in Kenya and Thailand. Swiss Airlines posts weekend travel specials on its website every Wednesday morning. Coop sells discounted flight coupons every year for travel to major European cities. There are often relatively affordable direct charter flights (Edelweiss is one company) to holiday destinations like Thailand and Egypt. Within Switzerland, definitely get a 1/2 Tax (Halbtax) card if you plan on using the trains at all. Buying your Zurich tram pass annually saves a lot of money.
Neighborhood: I lived in Kreis 1, as do many of my friends. It's good for single people who appreciate convenience and location. Kreis 2 is a bit quieter (and close to Google). Kreis 8 has a lot of expat couples and is convenient to the lake. Kreis 5 is the up-and-coming area for the Swiss equivalent of hipsters and techno lovers.
Most interesting restaurant experience: tie between Blinde Kuh (Seefeld), where it's completely dark, and Oepfelchammer (Niederdorf, ask for the Weinstube), where if you climb through the rafters and drink wine upside down, you can carve your name in the wall
Other Restaurants and Bars: Cafe Zaehringer in Niederdorf is reasonably priced, student-run coop that has good soups, veggie options, and is smoke-free on Sundays Aepli Bar in Niederdorf is one of my favorite bars in Zurich: cute, unpretentious, and campy in a Swiss way Zeughauskeller in Paradeplatz is the place to go for sausage and beer. Tearoom Blunt in Kreis 5 is a good place for Moroccan style brunch.
Lumiere on Widdergasse not only allows dogs (as do most restaurants) but has dog food and dog bowls, and will serve your dog dinner while you eat.
Best gummy candy:Baerenland (Marktgasse 11, near the Rathaus tramstop) has the best gummy candy in the world, and I consider myself to be something of a connaisseur. The German owner, Christian, is friendly and gives out more free samples than you can eat.
In the summer: they turn some of the outdoor swimming pools into awesome bars. The Frauenbad in the Limmat is my favorite bar in the world, and it's open to the public on Thursdays in summertime. Rimini (near the Boerse) does the same thing most nights in the summer, but I find it a bit more claustrophobic. Rimini stays open into the fall, because they put up a bunch of heated tents, which are pretty cool. If you can get tickets to see a movie at Orange Cinema in the summer, do it -- they set up an outdoor cinema and you watch movies on the lake, it's great. Go to Ikea and buy one of the little portable grills (they are round and look like a tiny UFO). Perfect for grabbing to take to the lake on lazy summer weekends.
Music: There are lots of summer music festivals in Switzerland, including the Montreux Jazz Festival and the Blue Balls Festival. Live music is great year-round, because the venues are often small, and the concerts are more intimate.
Other things to check out when they come around: Oktoberfest at the Bauschaenzli (if you can't make it to Munich, this is a fun local option); Sechseläuten parade and burning of the Böögg in April; parades and street celebrations all the time... Curling. There are thermal baths around Switzerland that are just awesome - Leukerbad and Vals are just two of them. There's a glass factory in Hergiswil near Lucerne that's cool, and the factory seconds store is deeply discounted. If you hear about a yodeling festival, go to it. They are awesome.