Oh, man.
Wow.
First of all, get everything in writing about the apartment. Do not go on the word of anyone, always get everything in writing. Smoking allowed? Has to be in writing. Paying a deposit or not, has to be in writing. Included costs? Has to be in writing.
2nd: talk to the
Zurich renters association. They can help you with the legalities. You have to pay a small fee to be a member, but really useful if you need it.
Next tip, don't move in any where without seeing the contract first. They can really rip you off otherwise.
My biggest piece of advice: get away from short-term contracts. If you can at all, try and get a long-term contract. I don't know your situation or how long you plan on staying here, but it really will help you save money and hassle.