You can use the public transport system buses, trams, trolleybuses and maxi-taxis or cabs. To get to Mamaia you can take the 301 mini van called maxi-taxi but it's very crowded.You can also take the special bus which takes you to mamaia and has regular stops around the city and resort,it is a double decker buss and you can find it as you exit the train station in the bus station. The best option is a taxi; a legal taxi must be painted yellow and must have the fee written with green characters on the driver's door . Always insist on starting the meter and try to avoid the cabs with red tariffs right next to the train station who ,as in every major city try to take advantage of tourists. Walk 100 meters from the station and only then wave to a cab. The typical fee from the train station to Mamaia is between 5-10 euros. You can also get to Mamaia by taking the 40 or 100 bus.
