Å vite = to know, to be aware of, to be certain, to recognize (in some contexts, interchangeable with "å skjønne" = to understand)
-jeg vet svaret = I know the answer
-vet/skjønner du hva du skal gjøre? = do you know/understand what you're supposed to do?
-jeg vet hvem han er = I know who he is
Å kunne = to be able to/manage, can, to know, may (in some contexts, interchangeable with "å klare" = to manage/be able to)
-kan jeg komme? = may I come?
-jeg kan å lese = I can read / I know how to read
-hvis hadde kunnet hopp
-jeg kan ingenting = I know nothing (jeg vet ingenting has the same translation but the meaning behind it may differ from context)
-jeg klarer ingenting = I'm not able to do anything
-jeg kan det utenat = I know it by heart
Å kjenne = to know, to sense/feel/notice (in some contexts, it's interchangeable with "å føle" = to feel)
-jeg føler/kjenner meg kvalm = I feel nauseous
-jeg kjenner ham = I know him (as in an acquaintance [en bekjent])
-jeg kjente en rar lukt = I sensed/felt [smelled] a weird smell