Geneva is divided into two parts -- the Rive Gauche and the Rive Droite (left and right banks) separated by the River Rhône and the lake. Most of the international organisations, and thus many of the international residents, are on the Rive Droite. This side also contains the airport and the main train station, although the second station, which serves many nearby French destinations, is at Eaux-Vives on the Rive Gauche.

On the Rive Gauche:

Plainpalais and the old town "Vielle Ville" are full of nice bars and restaurants. Due to their proximity to the University this is where many students meet up. A favorite, especially in Summer due to its huge terrace, is "Le Clemance" located on Place du bourg-du-four.

The Vieille Ville is  the place for a stroll in the centre of Geneva, provided your legs can stand the very steep hills. It's also filled with antiques shops, and the cathedral is there. For the best-value tourist attraction in Geneva, check out the archaeological dig under the cathedral -- some steps down through a gate just to right of the main entrance. Ask for an audio guide in English (free). The opening hours are a bit restricted, and it'll take you a minimum of an hour to see everything, so try to go a while before it closes.

Rue du Rhône and Rue du Marche are two of the main shopping streets and are located parallel to eachother. Rue du Rhône is Geneva's equivalent to Les Champs Elysees in Paris, with all the upscale boutiques. Whereas Rue du Marche which is a pedestrian street has all the other shops.

Eaux-Vives is also a mix of residential and shops/bars/restaurants.

Carouge, a bit further out, is different to the rest of Geneva. It's an Italianate city, much beloved of its inhabitants. Unlike the rest of Geneva, where the balconies tend to look out onto the streets, in Carouge the buildings are built around enclosed courtyards. Also a nice place for a stroll along the river Arve. 

On the Rive Droite: 

The other big department store is Manor, on the way up from the river to the station. All the buses go past it. Don't go in on Saturdays in December, however. 

Paquis is a lively/colorful neighborhood near Cornavin, Geneva's Main Train Station, which is both residential and filled with many shops and restaurants. It's open later than anywhere else in Geneva, though that isn't saying much.

Walk along the lake to the Paquis baths -- a safe place to bathe, provided you don't mind sharing with swans and you can bear to sunbathe on concrete. Carry on along the lake through a series of parks that mostly contain large houses now owned by international organisations. At the far end, you will walk through the grounds of the World Trade Organisation -- access is surprisingly open considering all the people who hate the WTO, although the grounds are closed off a couple of times a year for major meetings.

The path then takes you into a tunnel, under the rue de Lausanne and into the botanical gardens, where you will find pink flamingos and various ancient breeds of Swiss farm animals. The gardens are beautiful in spring and summer, just right to sit under a tree and read a book or walk around the "forgotten vegetables" garden.

Walking up the hill and to the left, you come out at the bottom of the grounds of the main United Nations building. You won't get in at this gate unless you have a pass. Turn right on the road for the Place des Nations, around which are many of the international organisations. The UNHCR has a visitors centre here. You still won't get into the U.N. here, though. Continue up the hill for the visitors entrance to the U.N. and, across the road, the Red Cross museum, which is more interesting but less pretty.

Also on the Rive Droite check out the lively life at Servette, and go shopping at the Balexert shopping centre, both a few minutes ride on buses from Cornavin station. Continue up the road beyond Balexert to the French border for the CERN nuclear research facility, which also does extensive and very interesting tours.