Zoltan, who holds a PhD in Philosophy, recommends that you visit The Garden of Philosophy.
There are a few very spectacular places in the Buda side, including the Castle Hill and Gellert Hill. If one visits the latter, the most obvious sights to see are the Citadella and the Liberty Statue. However, if you would like to visit a more peaceful place which is not so packed with tourists there are some more hidden and special spots on the hill.