Taking the train to Annemasse? I suppose you know that train leaves from the gare des Eaux-Vives, which is not linked to the Swiss railway system. To get there you will have to transfer twice (train from airport to Cornavin station, then tram #16 to 'Amandolier SNCF' station). Then it takes 10 minutes to downtown Annemasse.
Not too much to do in Annemasse, which is a rather down-to-earth town; however restaurants there have a much better price/quality ratio than across the border in Geneva. Some areas of Annemasse get a little sketchy late at night (idle youngsters mostly). If you are close enough to the border, walking over or taking a bus to the Moillesulaz border crossing will get you to the Geneva tram, which is more convenient than taking the SNCF commuter train to Eaux-Vives.
Most of my visitors cover Geneva within two days. The museums are not world-class, the international organisations are but office buildings, and shopping is expensive. Renting a bicycle and cruising around the lakeshore with a few ice-cream stops is pleasant in August.
There are nice day trips to be done on the northern shore of the lake (anywhere between Nyon and Vevey, minus the suburbs of Lausanne), either by train and/or boat. The Swiss railways have day passes which are a good deal for a long day trips to the Alps (e.g. Interlaken area, Zermatt, etc.)