It will be very helpful if someone spots the city centre of Zurich in Google Maps.
The center of Zurich is not very big, and most places in Zurich are actually whole streets and areas that are interesting. Generally you can consider the whole area between the Bahnhofstrasse, Bahhofbrücke, Seilergraben, Rämistrasse, and the Quaibrücke as the inner center of the city (more or less the district Kreis 1. There are some more places of course. Here some places to walk along:
The old city called Niederdorf, from Central to the Grossmünster and Bellevue, connected through the Niederdorfstrasse and then the Neustadtgasse. Alternatively (or on the way back) along the river on the LimmatQuai.
As a park area, the Utoquai, starting at Bellevue again and going along the lake until the Blatterwiese.
The central point of Zurich I'd consider to be the main station.
Other points that could be considered being the city centre:
- the old town with historic buildings
- the Bahnhofstrasse which is a famous shopping street
- and maybe Bellevue at the lakeside which is where a lot of events take place
All those places are pretty central.
For such geographical investigations, I'm mostly using wikimapia.org (could be described as a community of people drawing polygons on a satellite maps to describe what's depicted).
Disclaimer: I've never been to Zurich, so I can't tell how accurate is this perception of Zurich's city center.