If you like train rides, the one between Bergen and Oslo is very nice. Depending on when you go, it can be spring or fall on both ends, but at a few stations in the middle, snowmobiles will meet the train to pick up passengers going to small hotels. I didn't find it scary, but the ride basically consumes a day. I don't know how easy the logistics would be to take it only party of the way for the scenery and then return to Bergen. There's also a scenic train trip that's somewhat oriented toward tourists, but I've never done it. IIRC it mostly follows a river leading into a fjord and there could be places where the tracks hug the side of a steep hill overlooking the river.

To avoid scary trains, don't take the cog railway to the top of the hill in Bergen. However, you can walk or drive (I'm not sure whether there's a public bus) to the top for the impressive view over the old port and all on Bergen. Assuming it's still open, there's a concession at the top consisting of a not very appealing sit down restaurant and more of a fast food place. Hiking trails start here.

Within Bergen, don't miss the historical museum or the unique leprosy museum in an old convent which unfortunately keeps very limited hours. Walk along the old port and check out the fish market. The city has some pleasant parks and pedestrian zones. There are excellent restaurants specializing in new Norwegian cuisine, although unfortunately the wine selection is limited and expensive. Try to go to one of the places in a historic building, for example along the port.

Outside of town (doable by public bus) there's a very old wooden church. Farther outside, you should try to see some fjords. This is easiest by rental car, but pay attention to the ferry schedules to avoid long waits.

Depending on the time you have, you might want to consider doing a cruise along the coast to see the fjords (and go above the Article Circle to Lapland) by postal boat.