What is the best way to travel these countries?

From Prague to Bratislava is close and there are both buses and trains (they used to be in the same country until recently after all). For the rest, look at skyscanner.net and whichbudget.com for the low-cost airlines that might fly there (and it might give you an idea about the best order to go in).

Alternatively, if you are willing to spend many hours on buses, there are tons of international bus lines in CEE (Central-Eastern Europe).

Your choice depends on how flexible you are with time and how comfortable you are with making such arrangements in a foreign country. (For example, you can take Wizz from Budapest to Rome FCO and then another flight from Rome FCO to Bucharest, either adding a visit to Rome or not.)

As far as Burgas-Istanbul, I would suggest that you add a destination in Bulgaria too. You can take a bus (there are trains too, but the buses are much nicer and about the same price) from Burgas to Sofia (the capital) or Plovdiv (an even nicer town in the south of the country) from where there are also overnight buses to Istanbul (8hrs from Sofia, 6 from Plovdiv) super cheap. All of these lines run midnight buses which I like since it saves full days for sightseeing, etc. (Although I haven't lived there for 13 years, I am originally from Bulgaria and my boyfriend's family is from Istanbul so we have done this in both directions multiple times. :> )

My friends and I travel a ton and money is usually the first consideration, so I would not buy an Inter-rail pass which is unnecessarily expensive and restrictive.

And last, I would recommend joining couchsurfing.org where you can get very helpful advice specific to each place from people living there (like where to buy tickets, whether credit cards are accepted, etc.).

Definitely buy a pass on the Eurorail. It is the most convenient and you can easily get to any major city in Europe with it. Here is the website: http://www.raileurope.com/index.html

Do not buy each ticket separately, it will be too much of a pain. Good luck and enjoy! The most beautiful trip is going through Europe on train, you see every aspect of European life, I envy you for the trip of a lifetime you are about to experience :)