There are no shuttles from the airports to the hotel per the hotel’s site.
If you’re going to the Con hotel from Ronald Reagan National Airport a.k.a. “National”, a.k.a DCA, you should take the Metro Blue line to McPHERSON SQUARE Station, a 17-minute trip. You won’t have to change trains and the station is several blocks south of the hotel. From there it’s an easy walk or taxi/Uber ride. (Per Rome2Rio)
The fare is $2.40 off-peak and $2.75 peak, as of June 3.
From Dulles, you need to take a bus to get from the airport to a metro station.
The easier option is to take the Silver Line Express ($5 one way, purchased at their booth at the airport) which will drop you at Wiehle - Reston East, from which you can take the Silver Line train to McPherson Square. Off-peak fare is $3.85 and peak fare is $6.00.
You can save a little money by taking the Fairfax Connector route 981 or 983 between IAD and Wiehe-Reston East, which is only $2, but is much slower.
There is also the option of taking MetroBus 5A to Rosslyn and taking MetroRail from there to McPherson Square, but the 5A costs $7.50 and you still have to pay a $2.25 peak or $2 off-peak rail fare, so it is unlikely to be worth it.
When you arrive at McPherson Square, exit the station via the Vermont Avenue / White House exit. At the top of the escalator make a U-turn and cross I Street. You should see McPherson Square in front of you. Walk along the west side of the square up 15th Street and continue three blocks to M Street. Then turn right and you will be at the hotel.
The third airport is Baltimore Washington International or BWI. You need to take a shuttle bus from the terminal to the BWI rail station and then a train to Union Station in Washington. Amtrak saver fares are $5 but could be sold out, requiring fares as high as $35, though $11 would be more likely. MARC commuter trains ($7) are an option on weekdays, but are slower as they will make several stops. It is also possible to get off the train at New Carrollton (Maryland) and get the MetroRail Orange Line there, but this will be both slower and more expensive overall.
To get to the hotel from Union Station, take the Union Station to Georgetown Circulator bus. The bus stop is normally on the bus level of the parking garage (accessible from the mezzanine level of the station but, due to construction, has been temporarily relocated to picking up on H Street NE, outside the parking garage. Get off the bus at K St NW / 14th St. Walk back on K St NW to 14th St, then north on 14th St to M Street and turn left. (Just before that turn, you will find yourself on Vermont Ave. Don’t panic - that’s where you want to be.)
A taxi from Union Station to the Westin would cost roughly $15. Figure on $20 from DCA, at least $80 from IAD, and $125 from BWI. Uber and Lyft are also available.
The DC Metro uses the SmarTrip card, which you can buy at any Metro station. The card itself costs $2. You can “preload” it with money for fares. DC Metro fares are distance based, so you swipe the card when you enter the system and again when you leave. If you don’t have enough money on the card when attempting to leave, you’ll be directed to a machine that will allow you to add additional fare. It also appears that they have options for things like automatically reloading the card. There are also 1 day, 3 day, and 7 day unlimited passes you can put on the card (or app).
And I see that they have added an app: https://wmata.com/fares/MobilePay/SmarTrip-App.cfm
https://wmata.com/schedules/trip-planner/index.cfm helps in estimating fares.
There are a number of other transit systems in the region, but the only one of those of much interest to most visitors is the DC Circulator, which operates several useful bus routes, including service to/from Union Station and a route that circles the National Mall and Tidal Basin.
Washington DC has an extensive bikeshare system. Capital Bikeshare costs $2 (payable by credit card or their app) for a half hour rental. A 24 hour pass costs $8 and allows unlimited 30-minute rides. See their website for more details and suggested routes to ride. Closest bikeshare kiosk to the Westin is half a block away at Thomas Circle, 14th Street and Massachusetts Ave NW.
There are a number of other companies (e.g. Uber, Lime, and Bird) offering app-based bikes and scooters. However, many of those are problematic because they allow riders to just leave the bike or scooter in the middle of a sidewalk when they’re done with it. Capital Bikeshare kiosks are less hostile to pedestrians.
The basic layout of the city is in four quadrants, separated by two imaginary lines, running east-west and north-south through the U.S. Capitol building. Most visitors will spend almost all of their time in the northwest (NW), but there are some places of interest in each quadrant. At any rate, it is always important to know what quadrant something is in to avoid going far out of your way. Numbered streets run north-south with numbers starting from the Capitol, while letter streets run east-west. with A Street being closest to the Capitol. After running out of letters, the east-west streets use two syllable words (e.g. Euclid, Fairmont, etc.) and then three syllable names (Albemarle, Brandywine, ….) Streets named after each U.S. state overlay this grid and run diagonally. There are also circles, to complicate things. And there are numerous exceptions, but these basics should keep you from getting too lost.
Specifically for WashingCon, the con hotel is the Westin Washington DC City Center, which is at 1400 M Street NW. I will leave out the NW part in these directions. As the address implies, the hotel is on M Street between 14th St and 15th St. To complicate things, 14th St meets Vermont Ave at Thomas Circle, just east of the hotel and Massachusetts Ave crosses through Thomas Circle just north of M St. In practical terms, this means that you may have to walk around part of Thomas Circle to get to various destinations.
