Most of the major international airlines offer connecting flights from India to Washington DC.
Washington Dulles International Airport in Sterlin, Virginia is the primary airport to land reach DC. Another international airport is the Baltimore International Thurgood Marshall Airport, about 40 miles away in Baltimore, Maryland.
You can take a taxi for a getaway to a nearby city, though it can be heavy on your pocket. Driving on your own wouldn't be a great idea.
Amtrak provides great connectivity throughout the Northeast Corridor. You can travel from Maryland in Maryland Rail Commuter, or get a seat in Virginia Railway Express to reach DC from Virginia.
Buses can be used to commute from nearby cities to the District. There are many bus companies which provide this service.