My wife and I are spending four nights in Budapest at the end of October. We don't really know much about it or what to expect and would love some ideas as to what we can see and do while there.
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I have just come back from Budapest (last night!). I won't add too much more, Samanthas answer was pretty comprehensive. What I would suggest is, get a hop on, hop off bus/boat ticket and do that your 1st day.
You can cover off loads of the major sights, see them as you want, then pick up the next bus (they run every 1/2 hour or so). It will also give you some info about the city, and helps you get your bearings for the remaining days wandering about.
Personally, I dislike tour busses, but they do have their uses, and I found it good to do.
Also, go to this Italian resturant. It's amazing. Good cocktails, great food, nice staff. http://goo.gl/maps/CgcrX
I see the most turistic attractions have been mentioned. But if you want to know the deep history of Hungary and feel what happened in the past. You MUST check the House of Terror.
This might be a a bit bizarre. But i asure you it is a one of kind experience.