Budapest & ??? Where to spend another 3-5 days?

I would go for Eger, Bratislava and Vienna. Dubrovnik and Krakow are nice destinations but too far from Budapest.

I would second Ija's suggestions--especially Eger. You're right--it is charming. Lot's to see, but small enough to walk the city and feel cozy. Another suggestion in Hungary is Pecs. A beautiful city with lots to do--lots of history due to its location as well as great hiking in the surrounding hills.

Pecs, Eger, and Bratislava are all doable on day trips or nice overnights. I would commit 4 full days to Budapest, there is just so much to see and do there.

I would do Budapest for 4 days, Eger for 1 day and Krakow for 3. That's 8. Add a day to one of those three if you can do 9. It makes logistical sense where-as Dubrovnik really doesn't. You are welcome to browse our blog from last year which included all three of these places (and many others). Start here; http://lee-reid2010.blogspot.com/2010/07/epic-journey.html and read on for the next several entries.

Thank you so much for all the great information! What might be the best way to get between cities (say, from Budapest to Eger)? Would renting a car be advisable, or is it better to take public transportation?

I just wrote a 3 day itinerary on another thread: http://www.ricksteves.com/graffiti/helpline/index.cfm/rurl/topic/64619/prague-and-budapest.html It's a bit more comfortable if you do it in 4 days. It can also be extended by a river cruise on the Danube to a lovely little artistic town 20 kilometres North of Budapest called Szentendre. It has beautiful cobble stone streets lots of small churches, cafés, art galleries.

I'd chose Vienna over the other 2 you mention. Krakow was interesting but Vienna oozed culture, mystery and very much felt like the edge of the empire.
Vienna is a very easy drive from Budapest and I;d divide my time between them, but I'm a city girl. YMMV.

Vienna makes a good choice after Budapest - especially if you enjoy the history and architecture (a lot fo Vienna is neo - this and that). The Austro-hungarian history is fascinating. We did an easy city bike tour in both cities - if you do any cycling, it's a fun way to get more area covered than a walking tour. If you are looking for a restful stop, Lake Balaton in western Hungary is great - off the American tourist track, but very good tourist infastructure - Heviz has a wonderful hot springs pond. Cycling, ferries, walking, trains - you can get to Vienna from there, but you'd want to have the trains figured out in advance - there's not a lot of Engish spoken, though German is more prevalent.

The real Jewel of Eastern Europe is Lower Silesia, Southeast Poland. We have family there, in Wroclaw and Zlotoryja, which affords easy access to the unbeaten path. Two hour drive to Prague or Berlin but there is much to see and do in between.

Dubrovnik, croatia is astounding, dot miss it. Also some of the Croatian islands on way plus Solvenia ia amazing... you will love it. do the research.

If you like architecture, Szentendre outside Budapest has a fabulous open air folk architecture museum called Skanzen. We only had a few hours for it, but you could easily spend the whole day there. A little pre-planning is required, as you have to take a bus (either private or the museum's own) or a taxi from town to the museum. For both history and architecture, we really enjoyed our two day loop through Szentendre, Visograd (spending the night), and Esztergom. Rather than taking the train, we splurged on the hydrofoil back from Esztergom into Budapest, which was great. The Rick Steve's book says these towns aren't worth it without a car, but we did just fine with the bus. The buses were nice and kept very close to schedule. You can pay as you board and they give change. It does help to be able to say "I am going to ..." Sopron on the western border with Austria also has interesting history museums and architecture. It is definitely worth a fast half day or a more relaxed full day, if you are a history and/or architecture buff.

Oh, I forgot to mention Pecs. If you have an interest in Roman history it is worth a trip to the city just to see the early Christian tombs. The main church is also interesting to see how a mosque was turned back into a church after the Turks were driven out and there is another mosque in town that has recently returned to active use but you can still tour. One night in Pecs is sufficient, I think.

I would go to Vienna. It is like a mini Paris. While in Vienna, check out the opera at the Vienna State Opera House. It is well worth it experience. Bratislava is a short train ride (1 hr) from Vienna. Melk Abbey is also about an hour from Vienna.