Some form of Norway in a Nutshell is an excellent idea. I don't think there's a better way to see some magnificent fjords in just one day--while also getting yourself from Oslo to Bergen. Rick does a good job of laying out the options in his guidebook, but it isn't as straightforward as buying one train ticket or paying for one tour. It may be helpful to read Rick's material more than once, referring to a map as you go.
Oslo and Bergen both have a lot to recommend them, and you want to include Trondheim as well (also worthwhile), so you don't have a lot of time flexibility with only about ten days to work with.
There are trains between Oslo and Trondheim; you could also fly. While I assume there are buses, it would be a long trip, so I would definitely prefer the train if you want to use ground transportation.
There are no trains between Bergen and Trondheim; too many fjords in the way. I think you could patch together buses through really beautiful country, but it would take a lot of time, and Norwegian buses don't have the advantage of being cheap. In addition, you'd be using buses that don't run very often, which would be risky on a short trip.
I don't know whether there are non-stop flights between Bergen and Trondheim; you can check on skyscanner.com. I traveled between the two cities on the Hurtigruten coastal ferry. The northbound ships depart from Bergen at around 8 PM. They get to Alesund early the next morning then make a day-long detour into the highly scenic Geirangerfjord before stopping at Alesund again that evening. The ship than travels overnight to Trondheim, arriving in the morning (before 10 AM as I recall). So it takes 2 nights and a day to get to Trondheim from Bergen. The southbound ships are on a totally different schedule, I believe, and I don't know that they take the Geirangerfjord detour.
Since the ferry is a combination of transportation plus the equivalent of two hotel nights, it is not cheap, but it is an option. Fares are date-sensitive. It may be cheaper to travel from Trondheim to Bergen than from Bergen to Trondheim.
There's good hiking just outside central Oslo and Bergen. Spending one night in Flam (along the Nutshell route) would allow for some hiking in that area, or you could rent a bike and ride down the service road near the Flamsbana rail line (not cheap). The small town of Voss, also along the Nutshell route, is known for active-sport opportunities.
Edited to add:
I'd recommend checking multi-city fares to Oslo/Trondheim, Oslo/Bergen and Trondheim/Bergen, or vice versa, to see whether you have one or more options that won't be a great deal more expensive or involve a lot more flying time than a round trip to Oslo. It would be especially great if you could fly either into or out of Trondheim, because traveling there to and from Oslo/Bergen will take a lot of time (unless you fly) and will not be cheap.