At GVA, stay airside—enjoy efficient terminals and lounges rather than risking city transport delays.
Take the free airport train to Cornavin (7 minutes, free 80-min transit pass). Walk 10 minutes south to the Jardins Anglais and the Jet d'Eau. Continue along the lakefront promenade for 20 minutes. The walk is flat, and the views are immediate. Head back at the 90-minute mark to use your return free transit window.
Follow the lake walk, then climb to the Old Town in 15 minutes. The Cathedral of Saint-Pierre has free entry and rooftop views for CHF 5. For lunch, try Cafe du Bourg-de-Four on Place du Bourg-de-Four: mains around CHF 28 to 38. The UN Palais des Nations offers guided tours for CHF 15 if you book ahead.
With 10 hours, add a day trip to Chamonix by train (1 hour 50 minutes, CHF 35 return, runs hourly). Alternatively, the train to Lausanne takes 45 minutes and costs CHF 26 return: the old town and Olympic Museum are both excellent.