Well, "to climb up" is "monter" or "monter sur", while "to climb down" is "descendre" or "descendre de".
There is a difference in meaning. In the forms with a preposition, you aren't on the object before (or after) the action. While in the forms without the preposition there is the idea that you stay on the object but are moving up or down.
So you'll "monter sur une chaise" or "descendre d'une chaise" (you can't stay on the chair while moving up or down). For the hill or the stairways, you can say both, depending on what you are insisting (the movement -- the form without a preposition -- or the result -- the form with a preposition). So you'll rarely use "monter sur les escaliers" or "descendre des escaliers" as, excepted in describing children games or when you want to reach something, what is usually important is the movement, but for the hill both forms are commonly use and will suggest different mental images.