Have you already booked your flights? If you are interested in seeing the fjords from Bergen, it would make more sense to fly from Alesund back to Oslo, rather than go all the way back to Bergen. Here's my suggestions for your Norwegian itinerary:
7/1 - Arrive Oslo, spend 2 nights
7/3 - fly to Bergen, spend 2 nights
7/5 - pick up car, drive to Flåm or Aurland (3 hours), spend 2 nights
7/6 - spend the day in the Sognefjord, doing an abbreviated "Nutshell" trip - Take the boat from Flåm or Aurland to Gudvangen, then take bus to Voss, train to Myrdal, Flåmsbana cogwheel train back to Flåm
7/7 - drive to Lom via Solvorn & Jotunheimen National Park (4 hours), spend 1 night
7/8 - drive to Geiranger, in the Geirangerfjord (2 hours), spend 2 nights
7/9 - spend the day cruising the Geirangerfjord from Geiranger to Hellesylt and back
7/10 - in the morning, drive to Alesund, via the Trollstigen road (2 hours), then take an afternoon flight from Alesund to Oslo, spend last night at a hotel near Oslo airport.
7/11 - fly home
I recommend buying the Lonely Planet Norway book, which covers much more of Norway then Rick's books do. It will give you hotel & restaurant recommendations in each of these locations, as well as info. on local tours and sights.
I'd use www.skyscanner.com for booking the inter-Norwegian flights (Oslo-Bergen & Alesund-Oslo).
Use www.booking.com for hotel recommendations near the Oslo airport your last night, as well as for hotels in any of these other areas.
For rental cars, use www.autoeurope.com. I also recommend a good driving map, such as the Michellin Scandinavia map. You an also use google maps or www.viamichellin.com for looking at routes and driving times.
For details and schedules on your abbreviated Norway-in-a-Nutshell trip in the Sognefjord (Flåm or Aurland to to Gudvangen, bus to Voss, train to Myrdal, Flåmsbana cogwheel train to Flåm) see www.norwaynutshell.com/original-norway-in-a-nutshell/
You shouldn't have any problem booking these various segments once you are in the Sognefjord.
Hope that helps,