It is then High German classes with a Swiss twist, not Swiss German dialect classes.

http://www.integration.zh.ch/interne..._angebote.html
Angebotsdatenbank, Deutschkurse, your Gemeinde or Region

or if you want the city to help you pick one (other Gemeinde may have similar support offers, call them):
Integrationsförderung der Stadt Zürich
Stadthausquai 17
Postfach
8022 Zürich
Tel: 044 / 412 37 37

Additionally, RAV may pay for the course if you're a job seeker. No need to be unemployed yet RAV rules will apply, including some monthly number of job applications. Talk to your RAV advisor.

Urs Max, thank you so much for posting this link.

Oh, had I known this was out there, and all in English! http://www.integration.zh.ch/interne...n_english.html
and here in a bunch of other languages:
http://www.integration.zh.ch/interne..._sprachen.html

I'm sure many simply don't know that information is there... and I'll be passing on this link, for sure. A really big THANK YOU!

My pleasure

I wasn't aware of them either, until I did a quick search. I mean, I'm not really among their primary targets. And I do mean "quick". Though I'm a bit surprised by your answer as these offers are quite easy to find. I googled for "gratis sprachkurse für ausländer in zürich", doesn't seem to difficult, no?

Btw I also came across this one. The Kulturlegi may be interesting to those with a low budget as it gets you 30-70% rebate on offers in and around Zürich. The linked page also features a "free activities" guide containing a very broad selection of offers.