You can rent quality, almost new BMC road bikes from a company called
Bike Switzerland in Geneva. They are located just a few minutes from the main train station. Full disclosure: I do some guiding work for that company.
The Saleve is the closest big climb in the region (on the southern edge of Geneva), but to get into real Alpine pass riding then you'll have to go a bit further, with the best options probably being the Col de Colombière, south of Cluses, the Col de Ramaz, north of Taninges, or the Col du Joux Plane, north of Samoens, which have all featured in recent editions of the Tour de France. This year, the Tour will also visit the Col du Colombier, north of Culoz, but I don't know that one myself. There are also the Col de la Faucille and Col de la Givrine in the Jura mountains to the north of Geneva, but neither of those is as spectacular as the Alpine climbs, and those two roads are reasonably busy with traffic.
BTW - Hi BeSwiss, I'm also living in Lausanne, Switzerland. It's nice to see that there is at least one other person on this forum who is living here. If anyone needs to rent a road/racing bike in Lausanne, Switzerland then visit a shop called
The Bike, where I'm a mechanic.