In my personal opinion (and I'm speaking strictly from a tourist's point of view), I think the best mode of transportation to take from New York to Washington D.C., would undoubtedly be the high-speed super-efficient TRAIN (per ticket on business class costs just over US$100 which you can buy straight off from the vending machine! Cool!!).
That's me in the photo below at the historic Union Station... after a delightful train journey from New York and looking pretty well-rested (I hope!).
If you much prefer to take a plane, feel free to do so too....er, but my flight from Washington D.C. back to JFK Airport in NY was a veeeerrryy scary experience indeed! First of all, the planes used by American Airlines are much smaller (compared to Delta's) and as such, the flight is soooo bumpy (and not to mention, the many air pockets it hit along the way!!), I didn't dare to breathe until we had reached JFK Airport! Yes, I noticed some of the passengers onboard (and there weren't many) were deep in prayer too! Not kidding...
Moral of the story: I don't think I want to board an American Airlines plane ever again! And I called my travel agent in Singapore to ensure that she notes it down in my profile - pronto!
Cabs are really reasonable here in Washington D.C., especially if you're traveling within the same zone.
Alternatively, you may wish to take the efficient (and clean too - compared to NYC!) Metro/ subway. It's quite an experience, believe me. ;-) You'd also meet nice folks on these train journeys who are ever-willing to help make your trip here more stress-free....
Once you arrived at the Reagan airport, there is a train at the airport that will take you to downtown. It is a nice train ride through the town. If you lucky, you might see some of the high profile congressman riding on the same train.
This is a stop on trains to neighboring cities and a connection to subway. You can also buy souvenirs. Besides, this is nothing short than a monument of architecture.
I love the Metro and Union Station. The DC Metro is a model for what the rest of the US should aspire to.
If you travel via Amtrak to DC, you'll arrive at Union Staion.
Before you run out and grab a taxi, have a stroll around, it's a beautiful building on the inside and out.