Oh, if you're starting out with children in Zurich, I'd heartily recommend the so-called "Gemeinschaftszentrum", which means Community Centre. They are in several places around the city, and run by the City of Zurich in collaboration with "pro juventute" (the Organisation For the Youth). http://www.gz-zh.ch/
(here you can have a look at the centre in each area of the City)
It is in German, but the gist can be figured out with google translate. Also, if you just go there and talk to people.
Most Gemeinschaftszentren (pl. of Gemeinschaftszentrum, also just known a GZ, pronounced in German ghe-tsed) have an indoor and an outdoor play area, space for having a picnic, and often a rather basic café of sorts. No-one frowning at you if your children run around here ... that's the point.
There are offers for all ages, a slightly different range at each centre: e.g. play-groups, baby massage, parent-and-child gym classes, painting, dancing and children's flea-markets, children's discos. Some have an integrated library, others have woodworking workshops. Also special week-long activities during school holidays, or events related to the traditions of the season, such as painting Easter Eggs or baking Grittibänz on 6th December.
Most of these activities require a low fee, some are even free. Certainly hanging around to play, and to meet others, is free.
These centres are well-maintained, clean, in good shape, well-organised. But there is nothing flashy there, no major electronics or big screens, in other words, they are basic, solid, good and child-oriented.
They also are a good place to hire a hall, for anyone wanting to hold a private event, also not just for children. It could be that for such an arrangement there are restrictions on alcohol on the premises... not sure about that.