There is nothing better to do than escape the busy streets in Pest and have a nice walk in the woods. It is very easy to get to and gives the most amazing panoramic view of Budapest. There are benches and conkers, and it just feels and smells autumn. I love kicking the leaves and running around there.
Once you get tired of enjoying nature, you can walk back to Pest on Szabadsag-hid (Freedom Bridge) and have a nice meal or a cuppa either in Vaci utca or Raday utca. (I would personally go for Raday utca.)
Google map: bit.ly/PWdCXP
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The Citadel at the top gives superb views along the Danube.
I spent eight years on Gellerthegy and loved the location.