By plane:  You can fly to Constanța airport, located 25 km north-west of the city, near Mihail Kogalniceanu (see more info here).  From the airport, shuttle buses can take you to the city for less than 6.00 EUR (see for instance here).
More flights land in the Otopeni airport near Bucharest (see info here).  You can than take shuttles from the airport to Constanța, many companies offering relatively low fares, of about 30-50 EUR (see for instance: ALT, GAT, DT, etc.) and a convenient door-to-door service.
 

By train:  Constanța is connected by railway with all Romanian cities.  There are daily departures from Bucharest to Constanța (and return) all day, at different hours.  You can buy the train tickets online from the Romanian public railway company, CFR Calatori, or from European service providers (such as TrainLine Europe ).  A one-way ticket (Bucharest-Constanța-Bucharest) costs a bit less than 30 EUR.
 

By bus/car:  Reaching Constanța by car is convenient due to the motorway A2 (E81) connecting Bucharest and Constanța (the distance is about 220 km).  From the south, Constanța can be reached on E87, which crosses Bulgaria, through Varna and Burgas, to Turkey or Greece.  Typically, an international driver’s license is valid in Romania for 90 days from the date of entry.
 

Information about international bus routes can be found here.