It sounds to me as if you may not yet have a guidebook. This is a very, very costly area you're heading to. I wouldn't think of trying this trip without a guidebook.
I believe it would be difficult to find lodging in the fjord area for July/August, so you are probably facing the long, one-day trip from Oslo to Bergen.
I'd check flights back from Bergen as well as from Oslo to reduce travel time.
Rick lists his top sightseeing suggestions for some major destinations right on this website:
Oslo
Stockholm
Click on "At a Glance" for specifics.
I enjoyed Bergen a lot, and it has many sightseeing options (see below), but I feel as if many tourists would find 4 nights one more than necessary. On the other hand, Bergen is very hilly except in the area immediately around the harbor, so it takes more time and energy to walk around than you might expect.
Sightseeing:
RS lays out a walking tour in his guidebook. I haven't checked to see whether there's something on the RS Europe app.
Bryggen Walking Tour: 90 min., incl 20 min in Bryggens Museum, 20 min in Hanseatic Assembly Rooms (Hakonshallen), and 20 min. in Hanseatic Museum (unless still closed), rest outdoors in Bryggens district (many of whose wooden buildings are modern re-builds).
Bryggens Museum, big modern building beyond Radisson Blu Royal: Archaeology.
Hanseatic Museum: Old merchant house. Mostly closed but Schotstuene (reconstructed Assembly Rooms) has reopened with new exhibits.
Bryggen: Colorful Hanseatic quarter mobbed by cruisers 1000-1500.
Gamle Bergen (Old Bergen): Fifty 18C-20C buildings. Tours of museum interiors top of the hour included in entry fee. Seeing facades is free. Outside city center.
Sandviken: Hillside area (behind St. Mary’s Ch); cobbled streets, clapboard cottages housing designers’ studios, jewelry boutiques, cafes.
Floibanen (Mount Floyen funicular): City, island and fjord views. Hiking trails at top. T.O. has maps. Floien Folkerestaurant, top of funicular, is affordable cafeteria. 25-30 min walk back down to town (get Floyen Hiking Map when buying ticket); tough on knees.
Ulriken643 Cable Car: Ascends Bergen’s highest mountain. Great views, trails, restaurant. 40 min. back down. I think avid hikers can go up one lift then hike to the other.
Bergenhus Festning: Includes 13C Hakonshallen (ceremonial hall, reconstructed post-WWII) and Rosenkrantz Tower. Included tour provides good historical background on city.
Berhenhus Festningsmuseum (Fortress Museum): Norwegian military history and Nazi occupation.
Mariakirken (St. Mary’s Church), Dreggen 15: 12C with 15C frescoes and colorful pulpit.
Theta Museum, Enhjorningsgarden: Small museum about resistance movement in reconstructed room. Limited days and hours.
Lepramuseet (Leprosy Museum): Former hospital. Skimpy exhibit; need to borrow an English translation or take tour.
Marken and other streets between RR station and harbor: fine residential area.
Gestapomuseet, Chr. Michelsen’s Gate 10
Klosteret: Hilly area with wooden houses, S of Klosteret Sq.
Bergen Akvariet (Aquarium): well-presented; shark tunnel.
Kultur-Historisk Samlinger: Ethnographic museum. Was closed summer 2022.
Naturhistorisk Samlinger: Natural history museum
Outside town: rebuilt Fantoft stave church (take light rail toward airport then walk) and Grieg house.
The tourist office on the harbor (upstairs) was very helpful to me last summer. I imagine it could get really busy, though.
I'll address sightseeing in Gothenburg in a separate post.