I am not sure if there are free rinks, other than the Christmas one at the Altes museum but this is too crowded and small to learn.

There is a nice big one at Dolder that I have been to a few times. It is not free, but you can get an annual pass I) think. As far as lessons, no idea but I am sure you can check there. They also rent skates. It seemed reasonable price to me and the hours are decent and it is very large(Largest one in Europe according to the link) so you can always find a corner or something to get away from the crowd. And evenings aren't usually too crowded. Go weekend afternoons and kids will be skating circles around you.

http://www.wikifun.ch/wiki/Dolder_Kunsteisbahn
http://www.doldersports.com/en/20/default.aspx

Also, as an added bonus, you can cycle up to it, as the hill up to it is where I do hill repeats. It is a steep 2km climb at about 10%. Of course, the DolderBahn will get you there as well.

See, I manage to turn a skating discussion into a cycling discussion (magic huh).