Thé lake is lovely in Geneva - lots of boat trips on offer. You can take a boat up the lake to Nyon which is another nice town for a stroll. Also Montreux and/or Yverdon for longer trips on the lake. A walk around the lake and out to the Jet d'eau is nice too (back in Geneva now).
The old town (including St Pierre Cathedral) is worth pottering around but probably only takes half a day. There is a natural history museum that some like and some find a bit bizarre! Lots of posh shops between the lake and the Old Town if you're into that.
I recommend Annecy too and agree it's possible without a car but you could do more with one.
I'm a local
on the French side in the Jura mountains (also lovely but you'd definitely need a car!).