How much time will you have in total? I was just there for a weekend, but we managed to see Oslo, Bergen, and do the train journey. Here's a link to my
blog on it - there's 3 entries in total, so you just need to click 'next' two times

We were there in autumn/winter, so in June it'll be really different - no snow

- BUT....the fjords will be open for tours on, and they looked amazing! They only run May to September, so if you want to go out on a boat, you might want to start looking into it, as I understand it's much busier in Bergen at that time of year, and from what I saw, the best fjords are over there...
Sorry I can't help on the Norwegian family thing, and where they go.
In terms of stopping in any towns en-route on the train - we didn't, but that's not to say you can't. The train stops in a lot of places - they looked like mainly ski towns when I was there due to the snow, but they have other stuff in summer - I'm not 100% sure. Voss is quite close to Bergen, and it seemed the biggest we passed through, and on reading the guide, it seemed like a lot of adventure sports there - not sure how suitable for kids though.
If I went back, I'd go to Stavanger next, as it looks stunning there too - plus, mainly to stand on
this rock 
(south of Bergen - that's why I'm asking how much time you have, as you could fly to Stavanger from Oslo, and train up to Bergen then train back to Oslo....but then I've also heard Northern Norway is pretty nice too...)