If someone were to ask what to see first in Europe I would say
- Athens and Rome the birthplace of Western Civilization.
- England as the source of American culture.
- Paris for the late Middle Ages throuhg the Age of Revolutions.
- Budapest for an introduction to "the other" Europe.
But to answer your question. There are discount flights from Rome to Budapest, so depending on you time:
Rome
Discount flight to Prague
Train to Vienna
Train to Budapest
But thats very simplistic and depends on Budget and time. That could be a 2+ week trip if you just see the cities and a 3+ week trip with all the stops inbetween
If you only have 8 days then half in Rome and half in Budapest. The flight takes about 2 hours and costs under $100 if you pack light. Plan it well and you can see and do a lot.