Champéry is about 90 minutes by public transport from Geneva Airport, making it one of the easier Swiss ski resorts to reach without a car. Take the train from Geneva to Aigle, then the Mont-Blanc Express regional rail to Monthey, then a connecting train or bus to Champéry. The journey involves two or three changes but the trains run frequently and connect well.
By car from Geneva, the drive is around 90 minutes via the A9 motorway and the valley road up from Monthey. Note that Champéry itself is a pedestrian village during the ski season — you park at the village edge or in underground parking. In good conditions the mountain road stays open, but chains or winter tyres are essential in heavy snow.