There are two airports in Dobrogea:
The airport is connected to the city of Constanţa by some private bus lines.
There are 7 - 8 daily trains from Bucharest to Constanta (note that many of them leave from Bucharest's Obor train station, not from Gara de Nord). In summer there are direct trains to Constanta from most regions in Romania ( from cities like Cluj, Oradea, Brasov, Timisoara, Iasi etc).
The only direct train from Bucharest to Tulcea leaves from Bucharest Obor train station.
Check train timetables online .
The partially completed Bucharest - Constanta motorway has considerably improved travel times by car to this region.
Buses and minibuses from Bucharest to Constanta and the Black Sea resorts are very frequent; check online  for timetables. There are also buses from Istanbul to Constanta.
In summer there are direct buses to/from various cities in Romania, but also across the border to the resorts on the Bulgarian Black Sea coast and to Varna. Now, the motorway from Bucharest to Constanta is completed, you can get Constanta, by car, from Bucharest in about 2 hours .
Visit the dolphin show in Constanta.