I’ve heard there are several places to ice skate in Zürich, but the only one I saw so far was at the ETH.
For 6 CHF an adult could rent skates, and children could get theirs for 4 CHF.
You could also rent an Eistier (Ice animal) which may be used as an aid for skating, or as something to ride on if you get someone nice to push you around.
There was a warming hut with things to drink and eat, and for changing into and out of your skates.
If you wanted to see the twinkling lights of the night city, you came by after dark.
All this is in the past tense because as quickly as this rink was set up, it disappeared after a few weeks. Poof.