It is strongly recommended you use public transportation whenever possible while visiting DC to save both time and money. The city (and Silver Spring) is very pedestrian friendly and easy to navigate so long as you can read. If a car is necessary, Uber, Lyft and taxis are very common and reasonably priced.
The information below is the cheapest and easiest method to reach Silver Spring from each of the Washington area airports. Silver Spring is located on the Red Line of the DC Metro.

From Baltimore (BWI)

From the airport terminal, take the free airport shuttle to the MARC/Amtrak station. At the station, take the MARC Penn Line train south to Washington, DC (Union Station). You can purchase tickets at the

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station using a kiosk or the customer service desk.
**Be sure to take the MARC train and not an Amtrak train. Amtrak can charge up to 6-7 times more for the same trip!**
At Union Station, transfer to the Metro (underground) and take the Red Line north towards Glenmont and exit at Silver Spring station.
You can read about all the options to travel from BWI to Union Station here.
You can read more information about MARC here.

From Reagan National (DCA)

Reagan National is the airport closest to the city and is serviced by the DC Metro. Exit the airport terminal upstairs and take the Yellow Line north to Fort Totten or Greenbelt. Get off at Fort Totten and go upstairs to transfer to the Red Line. Take the Red Line towards Glenmont and exit at Silver Spring station.
Metro cards can be purchased at the station.
Read the detailed DCA to Silver Spring instructions, including a map, here.

From Dulles (IAD)

Dulles airport is not connected to the city via train (yet!), so expect to rent a car, take a taxi, Uber or Lyft, or use the rideshare service Supershuttle.
The drive will take a minimum of 45 minutes, depending on traffic. Plan for an hour.
You can plan all of your Metro trips here.