Although I'm a fan of rail travel, the journey between Budapest and Dubrovnik hardly seems to fit in with your requirement for an easy way to travel. The German Railways website, which enables you to plan journeys anywhere in Europe, quotes the total journey time as nearly 70 hours, involving many changes:
http://reiseauskunft.bahn.de/bin/query.exe/en
The easiest way to travel is to fly via Zagreb (or possibly Vienna). A typical journey, with a Malev flight to Zagreb and then a Croatia Airlines or Austrian Airlines flight to Dubrovnik has a total journey time of 4 hours 40 minutes. You can plan your journey using the planner on the Dubrovnik Airport website. (Don't worry that it only appears to offer flights
from Dubrovnik. When you enter your date of travel, it will show flights in
both directions):
http://www.airport-dubrovnik.hr/trazil.htm
Chris