Another post where I ask for recommendations

My wedding is fast approaching, and while I think Matthias and I have that under control (it'd be a bit late if we were still running around planning it, given the wedding is in two and a half weeks, after all!), we've only barely begun planning our honeymoon. All the flights and accommodation are booked, but we haven't yet started to plan what we actually want to do in the places we'll be visiting: Budapest, Vienna, and Prague. That's where you come in.

I had great success earlier this year asking my Dreamwidth circle for recommendations for things to do in Iceland when visiting with my mother -- people made fantastic suggestions, and the two of us were able to put together a good itinerary, and we had a fabulous time. Does anyone have similar suggestions to make for any of the three cities Matthias and I will be visiting?

Things we like:
- art galleries, museums, cool old buildings/architecture. We wouldn't want to spend the entire time doing nothing but visiting museums, but one or two in each city would be nice.
- walking, especially in quirky/pretty/interesting parts of cities we've never seen.
- good food and drink. He likes beer, but will probably have done investigations of his own and have that covered. I like coffee.

If anyone has knowledge regarding public transport (if there's some kind of 24-hour travel pass or the like, or if we need to pay on buses with exact change, or other local quirks to public transport systems), that would also be super helpful. In all three cities we'll be staying in hotels that are reasonably central. We will be in each place for roughly two full days and three nights.

Thanks in advance!