I have spent many a holiday over the last twenty five years camping in switzerland with my husband and kids (it's only a ten hour drive from UK). You'll absolutely love it. You can buy a visitors public transport pass valid for the length of your stay which entitles you to half price fares on nearly all public transport - that includes buses, trains, lake boats and the majority of cable cars and rack railways. The Swiss don't have it for themselves - only the visitors get it. You can buy it at any train, boat or bus station. As far as I'm aware, you can take bikes on all forms of transport. I have done much cycling there too - some really tough climbs, but the scenery is gorgeous. The Swiss are rightly very proud of their country, which is spotlessly clean. Are you camping or staying in accomodation? I can highly recommend a beautiful campsite on Lake Thun at a small place called Gwatt (south side of lake)with stunning views along the lake towards Eiger, Monch and Jungfrau. If you fancy a day off the bike a good day out is to take a bus from Gwatt to Thun - look around Thun (don't miss the castle), then take a lake boat to Interlaken (lunch on board), look around Interlaken and then get a train back to Thun - a lovely round trip. Anywhere in Switzerland is beautiful - you won't be disappointed. If you want a nice hotel on Lake Thun, look for the Niesenblick on the North shore at Oberhofen. It is the cheapest hotel on the lakeside with a nice restaurant too. Rooms look straight onto the lake. Sorry to go on - but this area is in the Bernese Oberland and is spectacularly beautiful with nice riding too. Some of the high passes may get a bit of snow in September - we have been turned back in august because a pass was closed due to a heavy snow fall - but it probably wouldn't last long. We may be there ourselves in September - in a box on four wheels though.
Enjoy.