There's something called the Bosphorus rail tunnel being built at the moment which means that for now Istanbul's stations are not reachable by train. From January until March there are trains while work is temporarily suspended but by the time you get to May it's a bus again. You need to get a bus from Istanbul Sirkeci station which will take you to the Bulgarian border. Basically you buy a train ticket from Istanbul to Bucharest but for part of the way you have to bus it.

The bus leaves at 10pm and then you rejoin the train at the border at around 4am! From Bucharest to Budapest it's a night train. Then Budapest to Vienna (a 3hr journey by day). Then a Eurocity train from Vienna to Prague which takes just under 5hrs (also a day train)

So all told, it's 2 night trains and 2 day trains so you have to allow 4 days of travel as part of your itinerary - it's possible to train it, so it all depends what you want to fit in as you could easily spend a week or 2 exploring each of these fascinating cities

There are at least 2-3 direct buses from Istanbul to Bucharest
- for instance at 1PM ( http://www.muratturism.ro/oferte-turism/transport-extern/transport-istanbul ) or 4.30-5PM ( http://www.autogari.ro/Transport/Istanbul-Bucuresti/14475-643.aspx?lang=en )
They get to Bucharest in 12-14 hours.

But it would be better to fly, it's way faster and not much more expensive (the direct buses, as well as the bus+train combination mentioned above, cost around €40 while plane tickets are around €45 with a budget airline if you book in advance).