What's the best way?
Mamaia is a suburb of Constanta, so you'll have to get to Constanta either by plane, train or (mini)bus and then use public transportation or a taxi.
By minibus it takes about 3 1/2 hours to get to Constanta. Minibuses leave from around Bucharest's Gara de Nord train station and arrive (comically) at Constanta's main train station. It costs about 20 -25 lei, see for timetables:
Even though it's slower, I'd suggest to use the train because it's much more comfortable. The best would be to take the InterCity train that leaves from Bucharest Gara de Nord station at 7.14 AM and reaches Constanta at 11.37 AM. You can check timetables at:
From the Constanta train station you can take a bus ( line 41, I think) or taxi to Mamaia .
From Bucharest to Constanta (Mamaia resort is a quarter of Constanta City) the best way to get is by... taxi ! Time to expect to do is 1h45min (75 % is a Highway on duty), price is $130-150.
That's rather expensive. By plane it takes just 40 minutes to get to Constanta and it costs about $ 135 two-way ticket.
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