From Flam you can take a boat to many different places along Sognefjorden.
See ferry info here: http://www.norled.no/en/maps-and-schedules/expressboat/sogn-and-fjordane/

The two common routes to Bergen are

1) (As you mentioned) Vangsnes – following National Cycle route 6. From Vikøyri there is a 1000m mountain pass to go over to reach Voss. There will be no campsites or shops for around 40-45km. But after that, you will find plenty of amenities (shops /campsites, etc) towards Voss and onwards to Bergen. At Stanghelle you will need to jump on a train (R40) to Trengereid to bypass a series of tunnels that are prohibited to cycle. It takes 19 mins and costs around 115kr including your bike. The R40 runs every 1 to 2 hours and it is usually easy to get your bike onboard (best to book in advance but usually not essential).

From Voss you could take road 13 to Hardanger fjord and cycle to Bergen from there without requiring a train. The route is also very scenic. See the Oslo – Bergen route for more info. http://www.cyclenorway.com/routes/norse-all-road-series/overview/route-1-2
http://www.ridewithgps.com/routes/42429436?privacy_code=AQr7GqkAmhpO32pF

2) Take the express boat from Flam to Rysjedalsvika Ferry Terminal. From there, you can take the eurovelo 1/ route 1 to Bergen.
http://www.cyclenorway.com/routes/atlantic-coast-route/overview/bergen-trondheim/
http://www.ridewithgps.com/routes/39267512?privacy_code=fYonryshba76jGZX

To get an overview of campsites in a specific area I recommend you download the app ‘Norcamp’ and use the map to see what is available along the route you choose.