We caught a train from New York Penn Station to Washington and stayed two nights at the Hyatt Regency Washington.
This is close to the Capitol Building so is a great location for walking to the Smithsonian Musuem, however we found that there weren't many restaurants in th area.
The best way to get around Washington is by Hop-on Hop-off bus, then walking everywhere.
Washington is a very beautiful city with wow moments around every corner, which surprised me. We loved the Washington Monument, thought the Lincoln Memorial was fantastic, and found the Smithsonian National Air and Space Museum very interactive - so a good choice for families.
Having only two nights meant that we only had one full day of sightseeing. I'd recommend staying a few extra days and making your trip more than just a day trip from New York.