OK - here goes....!
Oslo is a great city, we loved it there. We spend a day island hopping in the fjord. There's not much on any of the islands, but they're good to explore/wander around. There are plenty of spots for swimming (and lots of people doing just that), and one of them has a nice little pub.
The new national gallery was impressive. It's got a lot of paintings/artworks (inc The Scream, which is not in the Munch museum), but also a great design area which was really interesting. And on the very top floor, loads of accessible/interactive stuff which we loved.
The museums on the 'other' side of the harbour were also great - we went to Flam and Kon Tikki museums - both really interesting.
We had an Oslo card for a day, which covered all those museums (and more).
The Alter Bryggen (harbour) is full of restaurants etc - really buzzy and nice atmosphere. It's probably worth a meal there one night. If you go all the way to the end of the harbour, there's another swimming area and some sculptures and space to mooch around in.
The opera house is a must-see - allow time to walk up/over the roof and enjoy the view. It's really impressive.
The sculpture park is supposed to be really good, but we didn't have time to go there.
Are you travelling by train between the cities? We went as far as Myrdal on the train - it's a stunning journey.
Bergen is picture-postcard pretty. A really lovely place. The Bryggen area is obviously a must-see (make sure you walk through/behind it too, for a different view). There are some nice places to eat at the outdoor fish market - under the covered stalls. The menus may look simple (basically fish with potatoes/salad), but it's the freshest fish and totally delicious. DH had whale, which was nothing like we thought, but was lovely.
At the top of the funnicular are miles of walking trails. We hadn't realised how extensive it was, so didn't have the right shoes on for long walking. We did still walk a few miles (inc the walk down), but would have gone further in proper walking shoes. The views are amazing!
The Nordnes area is absolutely gorgeous - my favourite part of the city. Loads of lovely pretty streets of beautiful houses. It's worth an afternoon mooch (with a stop at a nice bar/coffee shop too).
Most of the 'things to do' seemed to centre on boat trips. We didn't do them because we had come in by boat from Sognefjord, so had already done lots of fjord boating, but if you haven't been on the water/to the fjords, then definitely do that.
It's a very compact and walkable city (I guess that may depend on where your AirBnB is!). Apparently it's the wettest city in Europe, so be prepared for that, but it was like being in the med when we were there.
Have a great time - and shout if you've got any specific questions!