Your best bet is probably to take a Greyhound Bus from New Carrollton, MD Metro Station
to Ocean City, MD (call 800.231.2222 or go to
https://www.greyhound.com/farefinder/step1.aspx to reserve tickets). You can take the Orange Line of the Metro to New Carrollton from DC. You can take a bus or the MARC train from BWI and other places in MD to New Carrolton. There is one bus a day which leaves
New Carrollton at 11:35AM and arrives in OC at 3PM. You don't need to
change buses, and the 3.5 hour time is not bad for a bus trip to OC.
Cost is between $30 and $60 dollars depending when you buy your ticket
and other options you select.
If you need to go at anther time,
try the Bayrunner Shuttle. To do this you will need to get to BWI
Airport. To get to BWI from DC you need to take a Penn Line MARC train. You can
get them at Union Station in downtown DC or from New Carrollton, MD
which is on the end of the Orange Line of the Metro (co-located
stations). Bayrunner Shuttle has regular service to OC with several
shuttles per day. Cost is about $100. You can get specifics at
http://www.bayrunnershuttle.com/east/default.asp Have a great trip