The situation in canton Zurich is a bit complicated because every transport company of the ZVV organisation has different rules. You can find the detailed table here in German.

If you transport the bicycle only inside the ZVV system, you have to buy your ticket a second time. If you have a half fare card (Halbtax), the bicycle only requires a half fare ticket, too. If you leave the ZVV territory during the bicycle transport, or if the second ticket costs more than 15 Fr. (10 Fr. with half fare), you should buy the normal train-compatible day pass for the bicycle, which is valid on ZVV lines, too.

You can transport your bicycle in the trams if there is enough space left for the other passengers (this applies for all ZVV member companies that allow bicycle transports). All ships on Lake Zurich except the small Limmatschiff allow bicycles. The Horgen-Meilen ferry accepts bicycles, too, it costs just 3 Fr., cyclist included. In double-decker trains (S-Bahn), bicycle transports are allowed except Mo-Fr 6-8am and 4-7pm.