We are looking for a quiet place to stay in Budapest that is not too far off the beaten track. What do you advise Buda or Pest and for what reasons?

Buda IS "off the beaten track". Some like it regardless -- it is quieter, leafier, can have a great view of the Pest waterfront....

Pest has the major museums, opera, concert halls, restaurants, 2 of the 3 train stations....and a FAR larger selection of hotels.

The Intercontinental on the Pest side gave us wonder views across the Danube. It was just a short walk to everything.

Without doubt one of our favourite destinations. You can easily get a quiet nights sleep on the Pest side. We stayed about 5 mins from the American Embassy and St Stephen's Basilica. District 5 I think it was. It was an apartment so nearby local shops were easy to find. There was a nice green area near the embassy that was good to walk through. Many sights were a short walk or short tram ride away.

If you have time, I recommend the trek up to the Citadel for some amazing views.