Ukraine to Romania

Playing around on Wikipedia's lists of destinations for each airport, there don't seem to be any flights from Kiev to either Bucharest or Cluj (the airport in Transylvania). However, with one plane change, you have lots of options. A connecting flight will be easier than a nonstop flight plus train, since the train from Budapest to Bucharest is at least 14 hours, and from Budapest to Cluj Napoca is over 8 hours.

I'd favor Lufthansa through Munich or LOT through Warsaw, but there are other choices. Just make sure it's all on one ticket. Use Skyscanner to see flight options: http://www.skyscanner.com/

Try and download the Inyourpocket guidebooks which normally have all the flight listings for the key destinations.

Also, check our Aerosvit and Ukrainian Airlines to see which destinations they serve.

The destinations you've mentioned are quite spread out (Kiev, Budapest, Bucharest, Belgrade, Sofia), and you'll be spending a lot of your time traveling. Remember to include all the travel time in your itinerary -- just moving around in this region is going to cost you multiple days. If you only have two weeks, you'll probably only get to see your top destinations (Kiev/Chernobyl, Bucharest/Transylvania). You may be able to work in a day or two elsewhere as you are en route between the two. If you have three weeks, you should be able to fit in your next top destination and perhaps a quick stopover at another. However, it all depends on how much time you want to devote to those secondary destinations. It's hard for me to say at this point without a more firm idea of your itinerary and how much time you'd like to spend at each location.

Are you interested in other parts of Ukraine or is your interest only in Chernobyl? Rather than try to cover so much distance on the map, you could include cities such as Lviv or Odessa. Direct domestic flights from Kiev would get you there much faster than the train, and it would give you an opportunity to see a little more of Ukraine. For what it's worth, I enjoyed Lviv more than Kiev -- Lviv is a little more like Europe, so it seemed more familiar to me. From Lviv, time permitting, you could continue on the train to Budapest, and then you could loop back into Romania or skip Hungary entirely and take the train directly into Romania. From Odessa, you could take the train into Moldova and then enter Romania from that side. With either of these scenarios you're no longer trying to make a huge circle to encompass so many far apart places. It just depends on how important your secondary destinations (Budapest, Belgrade, Sofia) are.

Just like you could do all flights from Kiev to Cluj (which looks fairly affordable), you could also do all trains, which takes about 26 hours with one connection (in eastern Hungary, rather than Budapest).
How to Look Up Train Schedules and Routes Online gives you the DB train schedule link and tips for using it.