Well, the first problem here is the fact that since winter Greek Railways suspended all the international trains and it doesn't look like the situation will be resolved any time soon, thus unfortunately you might need to cross the border from Bulgaria to Greece by some other mean of transport.
Concerning visas: both Bulgaria and Romania are member of EU, thus they tend to appropriate EU rules concerning traveler as well. As result NZ citizens (as well as US, Canada, Australia etc.) don't need visas. More info here -
http://www.mfa.bg/en/pages/view/5256 and
http://www.mae.ro/en/node/2040Concerning timetable: for Budapest-Romania you can easily use Hungarian railways website -
http://elvira.mav-start.hu/elvira.dll/xslms . Lately MAV-CFR cut the number of direct trains between the two capitals (Budapest and Bucuresti), but still there are a few of direct ones. In case of Romania-Bulgaria you can use timetable listed here -
http://bdz.bg/page.php?id=ext_razp or DB website offers online live version as well (
http://reiseauskunft.bahn.de/bin/query.exe/en?newrequest=yes&protocol=http:&)
Concerning tickets and best fares: choice of tickets actually depends on your plan, though whether you want to make multiple stops on the way for sightseeing or other things or you just one to do the whole trip with minimal stops like Budapest-Bucuresti, Bucuresti-Sofia and then go to Greece. If the later, then it makes sense to purchase discounted ticket Budapest-Bucuresti (seats from 29 EUR, can be bought online from Hungarian Railways website. NB need to buy them in advance since the number of tickets for such discounted fare is limited) and then already in Bucuresti ticket till Sofia (around 25 EUR). From Sofia catch a bus to Thessaloniki (Sofia bus station info
www.centralnaavtogara.bg) and then continue to Athens by Greek domestic train from there (fares and timetable on our Greece page -
http://europetrainsguide.com/Countries/Greece/Greece-OSE.html). If you do plan to make stops in Romania and/or Bulgaria (besides obvious Bucuresti and Sofia for train changes), then it might make sense to look at 'tricky' tickets such as CityStar MAV-CFR-BDZ.