We did some research for you, Rebecka. These are our suggestions to stay fit in Zurich:

You’re lucky, one of the nicest gyms in Zurich – Asia Spa – has a summer special on offer: Get a one months fitness and wellness subscription for 199 swiss francs!
Asia Spa is at the shopping centre Sihlcity, it’s easy to get there by public transport (Trams Nr. 5 & 13).

There also are a few Migros Fitnessparks in town to choose from. Buy a subscription for 10 entries (365.-) and use the whole wellness and fitness area of all the centres in Zurich (exept the Hamam at Niederdorf). Another option is to get a bodymind summercard (380.-) for 3 months.