There is an excellent website providing good information on travelling with a dog on public transport: Travelling with dogs | SBB
Generally said, you can transport small dogs with a wither height of up to 30 cm for free as hand luggage provided they are in a transport box, basket or other animal-friendly container and placed under your seat. However, once the dog is no longer in its container, you will require a separate ticket for the dog.
Dogs that need a ticket, need either a ½ fare point-to-point ticket, or an Unireso Day Pass zone 10 at reduced fare (valid in Geneva), or an Unireso 9 Day Pass Zone 10 valid from 9 AM at reduced fare (valid in Geneva), or Day Pass Dog (valid in the entire country), whatever is the cheapest for the planned journey(s). It also applies for travelling on the Mouettes.
A 1-year Dog Pass is, in my opinion, a viable option provided you are staying for multiple days in Switzerland. It does require / comes on the SwissPass card, while the products mentioned above don’t.
unireso transport ticket fares | tpg
Travelling with your dog | SBB
If that all sounds too confusing to you, you may wish to see the SBB Travel Center at Geneva main station (locals call it Cornavin station). There, they can help you to choose the best ticket.