I've only made one trip to Norway (this year, and carless). There are others with considerably more experience than I have, but perhaps I can get you started.
The Naeroyfjord is a branch of the Sognefjord. It's quite narrow (thus its name) and is considered one of the two most beautiful fjords in the country. The narrowest fjords tend to be the most spectacular. The other fjord most often mentioned is the Geirangerfjord. I don't think you can sleep in the area around the Sognefjord and take a day trip to the Geirangerfjord, even with a car. It's just too far away.
Public transportation in rural Norway doesn't run very frequently, so if you don't have a car, it will be a challenge to range very far from your base on any given day. I don't remember whether this is something mentioned in the RS guidebook or is just a comment from one of the RS staffers here, but it was suggested that if you only have one night to spend in the area, Flam (or perhaps Aurland) would be the most practical base. They're on the south side of the Sognefjord. Balestrand was recommended for those with a bit more time. I believe more than one of our posters has stayed in Balestrand and really liked it, but I'm not sure any of those (few) folks were depending on public transportation. Balestrand is on the north side of the Sognefjord.
The Flamsbana scenic train, the Naerofjord ferry and the easiest bus route to the Borgund stave church all depart from the Flam side of the Sognefjord (actually, they all depart from Flam itself). If you stay on the other side of the Sognefjord, any of those excursions will require you to find a way across that large fjord twice. The crossing method I'm aware of is the express boat to/from Bergen, but I think it goes to Flam just once a day, so it's really not a very practical solution for those with sightseeing plans on the south side of the Sognefjord and no car.
The above situation is what led me to discard Balestrand as an option, though I started out wanting to stay there. I considered Aurland (less touristy than Flam) but I was unable to find information about bus service between Flam and Aurland. In the end, I came to the conclusion that Flam was best for me even though I had two nights to spend in the area.
I didn't really explore bus options from Balestrand north, beyond discovering that the Geirangerfjord was way too far away. There might be some nice, scenic bus routes from Balestrand, but everything I looked at on that side of the fjord seemed to run only once or twice a day.
Weather in fjord country and western Norway is really iffy in terms of precipitation. It will be valuable to have some extra time in that area so you can try to match your activities to the weather.
Bergen is an extremely scenic city with lovely architecture. It's worth some time.
I decided to travel north to Alesund (known for Art Nouveau architecture) on the Hurtigruten ferry, which departs Bergen at 8 PM--5 times a week at this point, I think, but there's another company offering service as well. After making a stop in Alesund around 8 AM, the ferry detours into the highly scenic Geirangerfjord and makes a second stop at Alesund at 8 PM. So you can see the Geirangerfjord from Bergen with an overnight on the ship followed by a night in Alesund. I think there are other options for seeing the Geirangerfjord from Alesund--I'm not sure whether by ferry or bus/ferry combination.
One of the best tips I read about visiting the fjords was on this forum (don't remember whose comment it was): Many people find the fjords more impressive when seen from above (i.e., from a car or bus on a road above fjord level) than from a ferry down on the water). Most of my fjord views were from the water; they were lovely and worthwhile, but it would have been nice to get the bird's-eye view as well. I think there's a bus tour from Flam/Aurland that takes you to a viewpoint.