You need to go the MÉDECINE TROPICALE ET HUMANITAIRE department at HUG (https://www.hug-ge.ch/medecine-tropicale-humanitaire). You can go without an appointment, just check their opening times. Once you go there, you get a number, they'll give you a form to fill in, when it's your turn, a doctor will sit and talk to you for around 20 minutes to decide the vaccines you need. Then you'll go to another doctor (in the same department) to receive the actual vaccines. I paid around 250 CHF, but the price may depend on the number and type of vaccines, but it will be in that range.

I think any doctor will test you for zika. But they just send the blood test to a laboratory, so you end up paying more. I think you can go directly to some of the laboratories, e.g. Unilabs. According to their site, they seem to charge CJF180 to analyse for zika, (That probably does not include the price for taking the blood sample.)

It will probably be cheaper in France, you can try Oriade Noviale in Ferney +33 4 50 40 57 51

You're a star, thank you!