So, Zurich may not be the most appealing city in Switzerland and January not the greatest time to go, but we have been given tickets and a place to stay (apt near the main train station) AND, a great reason to go - to see our dear exchange son get married. What else should we do while in Zurick (lv 1/19, return 1/28, wedding is on 24th) we are considering the possibility of maybe one overnight trip somewhere. (funds are pretty limited)
Zurich is considered to be one of the most "livable" cities in the world, but it's tourist attractions will take no more than a day to get through. There are plenty of day trips you can take from Zurich. The possibilities are endless, get a copy of Rick's Switzerland guidebook for all your options. Top on my list would Luzern, Jungfrau/Schiltorn, Luscanne, and one of the scenic train journeys like the Glacier Express. The Swiss Pass will save you a lot of money on trains, buses, trams, boats, cable cars, and free entry to most museums.
If you ski, welcome to paradise on Earth. If not, you can still take the cable cars to the mountain peaks for some spectacular vistas. You can't get really good Alpine scenery from Zurich -you can technically see the mountains on clear days from various points of the city, but the views aren't great.
One problem, though... if funds are limited, you will need to budget everything very carefully, because Switzerland is one of the most (if not THE most) expensive countries in the world.
Although Rick Steves' Switzerland book is pretty good, just be aware that budget travelers are not his primary focus. You may want to check out The Rough Guide, Lonely Planet or even Let's Go.
Sometimes it is just nice to spend time in a city, walking the neighborhoods, visiting the stores, churches, museums, galleries, etc. Travel does not have to be about seeing all the "must sees" and then quickly moving on. We spent 4 days in Paris and spent most of it simply walking around. Yes, we went into Notre Dame, St. Chapelle, Sacre Couer, etc.(yeah, I can't spell French) but we did not go up in the Eiffel tower, nor in any museums or galleries. We came home satisfied though and with plans to go back.
One finds the gems of a city by sticking around for a while. I bet there is plenty to see in Zurich for a whole week. I am sure that your exchange son will have plenty planned for you to show off his city. Perhaps a local side trip to some near by towns since you are short on travel cash.
Having seen Zurich, I would take the short train ride to Interlaken and then the cable car up one of the mountains for a real view if the Alps.
I found nothing of interest in Zurich to see.