Here are a couple of maps about the Old Town in Zürich.
This online map wil help, as well.
The part near the Hauptbahnhof (main station) is easy to get to, has a lot of interesting walking between Paradeplatz, St.Peters, and the Lindenhof. There is some expensive shopping at some very very fashionable stores and boutiques near Hotel zum Sörchen and restaurant Zeughauskeller, and is the most common tourist area.
The area on the other bank of the river, in the Niederdorf is less often visited by tourists, but has some really nice old buildings, a lot of history, and lots of small shops and stores. I stay near there whenever I am in Zürich.
I discussed this in a bit more detail in another thread, with a bit of information about the public transport network.
Look in particular at the 7th post in that thread.
I have attached a couple of images from September 2010. The photo of the Heimatwerk store ( Swiss handicrafts) shows a walkway just south of Uraniastrasse along the west bank of the river which gives nice views to the other side. Further south, the twin spires of the Grossmünster are nicely visible. Above the walk on the west side is the Lindenhof, a hill overlooking the river, at one time a Roman fort or camp.
Arno mentioned Uetliberg. Its a nice place to go up to for views. Take a train from the main station (twice per hour , the S10 train), get off about 27 minutes later at the end of the line at station Uetliberg. There are some nice hiking trails along the ridge from there, with restaurants along the way, but you should go up the hill to the Hotel/Restaurant UtoKulm, where there is a viewing tower. In the attached picture, taken by a friend of mine, the viewing tower is on the left. The taller one on the right is a TV tower.
The possible hike to Felsenegg with self service coin operated cable car back down to the train station at Adliswil and return to Zürich at is quite nice, but uses up about 2 to 2 1/2 hours that you may not want to spend up there. I often do that for a jet-lag afternoon when I come into Zürich from the USA.
There will almost surely be some snow up there at the beginning of your trip, so that walk might not be accessible.