Swiss destinations are almost all served by the CFF from the central train station Gare Cornavin while ski resorts in the French alps and the Jura can be reached by bus from the central bus station off of Rue de Mont Blanc or from SNCF's Gare des Eaux Vives. The price of the bus ticket often covers ski lift tickets as well, be sure to ask.
Here are just a few places which make a good day trip from Geneva:
Lausanne
Montreux
Gruyeres
Chamonix
Evian
Annecy
Yvoire
Thonon
by hitchhiking
Hitchhiking is relatively safe and more common in Switzerland than France, for example. The A1 motorway surrounds the city, with connections to the rest of Switzerland and neighbouring France.
To hitchhike to the direction of Lausanne North take bus number 29 towards Gare Zimeysa and step out at stop Blandonnet. Walk back 200m Route de Meyrin towards the center, across the bridge over the highway and you´ll find an on-ramp to highway towards Lausanne. Another, even better possibility is to take tram 14 or 16 in direction of Meyrin and step out at the Avanchet. Then walk forward 200 m. Walk down 100m along the on-ramp and hitchhike before the speed gets high. The position is very good, speed of the cars low, visibility good and there´s plenty of space for cars to stop. You should accept a ride at least to Nyon, where you can continue hitchhiking on the on-ramp. Hitchhiking on the on-ramp is illegal. Your best bet is usually to try and get a ride at one of the gas station/restaurants on the autoroute itself.
To hitchhike to the direction of Chamonix and Turin South-East take bus 27 towards Thônex-Vallard-Douane and go to the end of the line. Walk through customs to France and stand at the end of the customs just before the cars speed up for the highway. Be sure to have your passport with you when crossing the border. The position is very good, the customs officers are nice, speed is low, there´s space for cars to stop, all the traffic is passing through.
To hitchhike to the direction of Lyon and Paris South-West, West, North-West take the bus 29 to stop Blandonnet. Walk about 600m to the next on-ramp in direction of South, the one leading to the highway in the direction of South from Route de Vernier. The position is not very good because the cars speed up and visibility is not really good but there´s place for cars to stop. Take a ride at least 10km South to the Swiss-French border, where there´s a decent spot to continue. Walk through the customs and hitchhike - preferably with a sign - before the cars speed up. There´s not much space for cars to stop but they can, speed is low and all the traffic is passing through the customs.