As a UK English speaker, I do not think there is a BIG difference between "I climbed up the stairs" or "I walked up the stairs". It’s a matter of emphasis. You can also say “I went up the stairs”.
If you just mean you are going to the upper floor of a house, it is most common just to say "I went upstairs". In this case, upstairs is a place: “the bedroom is upstairs”. So "I went upstairs" just means I went to that place. The route taken and the method are unspecified. You could say "the bird flew upstairs", but it could clearly not fly up THE stairs.
“I went up the stairs” emphasises the stairs as the route taken: "I went up the stairs because the lift is broken"; "I cannot fly, so I went up the stairs".
"I walked up the stairs" emphasises walking as the method used to get upstairs "the old woman walked up the stairs instead of using her stair-lift". “She has trouble walking up stairs due to arthritis”. It might also emphasis both the method and the route “The lift is broken and I am exhausted because I had to walk up the stairs”.
"He climbed the stairs" is something you might find in older books simply meaning he went upstairs, but nowadays it implies some abnormal effort "the baby climbed the stairs today without any help”; “though shot in the leg he nevertheless managed to climb the stairs”.