The obvious things — Munch, the National Museum, Vigeland — are genuinely good, and the non-obvious part is how fast the city turns into forest and fjord at the end of a metro line.
Vigeland Sculpture Park
Gustav Vigeland's sculptures fill Frogner Park; the park is always open and costs nothing. The Vigeland Museum at Nobels gate 32, the studio where he worked, keeps normal museum hours and charges admission.
MUNCH
Eleven floors of Edvard Munch at Edvard Munchs plass 1 in Bjørvika, with four permanent hangs and rotating shows. Sun–Tue 10:00–18:00, Wed–Sat 10:00–21:00; the late evenings are noticeably quieter.
The National Museum
Norway's largest collection of art, architecture and design, in the purpose-built museum by the harbour just west of the City Hall. It is big enough that half a day is realistic; check hours before you go.
The Fram Museum
Bygdøynesveien 39 on the Bygdøy peninsula: you walk aboard the polar ship Fram itself, with Amundsen's Gjøa in the next hall. 180 kr adults, 70 kr children and students; 09:30–18:00 May–September, 10:00–17:00 October–April.
Holmenkollen ski jump and Ski Museum
The world's oldest ski museum, open since 1923, sits under the jump; one ticket covers the museum and the tower at the top of the in-run, which is the view people actually come for. There is also a ski simulator and a disc golf course on the hill.