Everyone has their own idea of "best" but I love small villages, mountain scenery, hiking and castles, so I would choose among these:
1) Bettmeralp, a car-free village high above the Rhone valley, with cablecar rides or walks up to see the spectacular Aletschgletscher. Wait until 26 June or the following weekend and check webcams to see how the weather and snow conditions are.
www.bettmeralp.ch Take the train to the town of Betten (2.5 hours from Geneva) and ride the cablecar up from there. We like Hotel Panorama.
2) Gstaad is a high-end ski resort but very pleasant in the "off" season, like now. The train takes you on part of the scenic "Golden Pass" route. The town has a nice pedestrian zone and is surrounded with lifts for alpine walking, as well as bike paths. The nearby town of Chateau d'Oex is a center for hot-air ballooning. Posthotel Rössli is an olde, traditional and very friendly hotel a short walk from the train station. We had a large triple room with a HUGE balcony. The garden restaurant is lovely.
3) A closer option would be the small town of Vevey, just past Montreux right on the lake; less than an hour by train. I believe there is a weekend market, and it is walking distance to Chateau de Chillon, a must-see. You could also ride up to the alpine scenery of Roche de Nays:
And I believe there are vineyard walks in the area as well.