If you are arriving by plane you will land at one of 4 airports:
City of Derry Airport (LDY) has direct flights to Edinburgh; Glasgow; Liverpool: London Stanstead. The airport is approximately 9 miles from the city centre. There is no train connection but there are frequent buses to the city centre (which take around 45 minutes) and taxis are available (which take around 15 minutes).
Belfast International Airport (BFS) has many flight connections to GB regional airports and also to Europe. You can take a bus from the airport (Service 109a/109b) to Antrim Railway Station and from there take a train to Derry/Londonderry.
Alternatively you can take a bus (Service 300) to Belfast city centre and then take either a train or a coach (Service 212) to Derry/Londonderry. The coach and rail stations are in the same building.
Belfast City Airport (BHD) connects to a large number of regional airports throughout GB and also has some European connections. It is only 15 minutes from Belfast city centre. Take a bus (Service 600) or taxi to Europa Bus Station and from there you can take a coach (Service 212) or train to Derry/Londonderry. The coach and rail stations are in the same building.
Dublin Airport (DUB) offers the widest selection of routes across Europe and especially from the USA. There is no train connection to Dublin Airport. There is a coach service (service X3 or X4) to Derry/Londonderry which departs from Zone 11 outside Terminal 1.
There is a more frequent (24 hour) coach service to Belfast (service X1 or X2). which may be helpful, depending on your arrival time.