The old part is definitely something you should see. There are a lot of small roads that lead you into nice little backyards where you don't feel like you are in a city at all. There you will find cool little shops (cafés, bars, and vinyl stores!!) that you should check out as well.
When the weather is nice, you must go to the lake! It is simply beautiful, and you will meet tons of people there. A nice spot to go is the Rentenanstalt Wiese. It is always very crowded, so if you are looking for a quieter place, take a tram to Bahnhof Wolishofen and cross the road, where it is a lot quieter. The river Limmat is also a very popular place to go! From the main train station, simply walk towards the national museum, through the park, along the river, and cross the bridge! It is called Lettebadi, and it is simply great! You will see people swimming in the river until midnight and even later! The bar/restaurant is open during the whole evening and until late summer.
Quick Tips:
To find out about local events or attractions, a visit at the local tourist office makes a lot of sense. They can inform you about the events that are about to take place, special concerts, etc.
Best Way To Get Around:
The trams are cool! And they are really convenient to get around with. Buy a day ticket if you plan to use the tram often.
(The Zurich Card might be an option for you if you are in town for 24 or 72 hours.)
Taxis are quite expensive, and actually, people just take the tram, not the cab; otherwise, the buses are good to use too, and they are actually like trams. They come very frequently, etc.