Hi Andrew,

I'm not sure about hiking places in Valais, Bern or Fribourg, but I know one in Simplon Pass, about 3 hours from Geneva (depending on traffic and whether it is school holidays) and it borders with Italy. We found it by accident on the way back from Lake Maggiore.

Here are some pics to entice your hiking appetite.

I can always recommend going to Sixt-Fer-a-Cheval 

Those are definitelly some nice places. I forgot to mention though that I don't have a car. So I can only go to places that have train or train/bus access from Geneva. Incidently they are both near to some of the destination I have planned for myself. I wonder if I can reach them by some bus from those places.

Try this:

http://fahrplan.sbb.ch/bin/query.exe/en?seqnr=1&ident=lk.027559190.1351252305&HWAI=JS!ajax=yes!&HWAI=CONNECTION$C0-0!id=C0-0!HwaiConId=C0-0!HwaiDetailStatus=details!HwaiConnectionNumber=1!HwaiMoreDetailStatus=none!HwaiAdditionalInformation=none!;~CONNECTION$C0-0!HwaiMapStatus=UNDEFINED!HwaiMapNumber=UNDEFINED!HwaiMapSlider=UNDEFINED!;

The public transport should be able to connect you from geneva to Simplon Dorf.. you need to take a train to stop at Brig, walk a bit to the bus stop and take the bus.

And before you go, it might be a good idea to check with these guys too: http://www.wanderland.ch/en/routen_etappen.cfm?id=317998 and ask for the hiking conditions, access, etc..

Good luck!!

For one day trips, espeically as the days are shorter, the best bet is Haute Savoie. There is a book by Janette Norton called Walking in the Haute Savoie which is a good introduction.  Olenty of other books available.  Otherwise join a club.

To go as far as Simplon for a one day walk will mean travelling in the dark or having a very short walk

Just seen you have no car.  It is possible to get around in Haute Savoie but destinations are limited eg Chamonix, Passy, Morzine. check the bus station in Geneva or google autocars haute savoie

But joining a club where car sharing occurs is the best bet unless you are desperate ti hike on your own

You could also consider renting a car or signing up for Mobility car sharing.