I don't know the Alps around Geneva that well but there is plenty to do!
First, from Geneva you have the option of staying in France or Switzerland (CH). France is obviously cheaper. And if you stay in Ferney-Voltaire or Annemasse in France you have public transport into Geneva.
You have access to both the Alps or the Jura mountains. Jura is lovely for hiking in the summer, but are not as spectacular as the Alps. Having said that you have a great view across to the Alps.
You can travel to Chamonix reasonably easily.
In CH, the Canton (countuy) of Valais (Wallis) is easily accissible by train from Geneva in a couple of hours and there is mountain galore! More than can be put in this post. Obvious places to visit include Zermatt (hike up to see the Matterhorn), Saas Fe, Leukerbad (hot springs) and impressive mountains (including some hard Via Feratta), Aletsch glacier. All are reachable by public transport from Geneva but if you are planning on doing longer walks you might want to look into staying in a small hotel, b&b or camp site.
All of Switzerland can be reached by rail from Geneva so check the forecast! Often you will have great weather on one side of the alps and lousy weather on the other so you can usually hunt out good weather.
In Geneva, the old town is worht visiting. The University park with the statues of the reformers (fathers of pretestantism if you are interested in that sort of thing) is worth a visit. A stroll around the lake is nice. Take a trip with the paddle steamer on lake Geneva (or one of the other lakes in CH: Lake Luzern is very very nice!). Take boat to Yvoire in France (old town with some good food, not too big so easily visited in a day). You should sample the local fish if you have a chance (filet de perche). Very yum! Oh and thepub Brasseur (Brewer) at the train station makes cracking beer and food to go with it. A 5L tower of beer outside in the summer is hard to beat!
OK, that's a quick over-view. If you want more details on anything let me know and I can suggest more places to visit in CH or more details
Edited: 09/05/2012 at 12:05