I would suggest early fall for a trip to DC. It will still be warm but the crowds will be lower as schools are back in session- and it's too early in the semester for field trips! Cherry blossom time is beautiful but the crowds are high.
To get into a White House tour, you need to contact your state representative or senator, about 6 months in advance. They are the only ones that can arrange the tour. For the Capitol Building, you can take the public tour or arrange a private tour through your senator/representative. I like the private tours better; they are usually conducted by an intern from your representative's/senator's office, and they are full of interesting information and anecdotes.
If you are flying into DC, fly into Reagan/National airport, NOT Dulles. It's difficult to get into the city from Dulles. Another option is to fly into Baltimore and take the train to DC. There is a shuttle from the airport to the train station, both Amtrac and MARC (cheaper, area commuter rail) run from the same station at BWI into Union Station in DC. Definitely use the subway while you are in DC; it's very easy to follow.
I highly recommend taking the Monuments by Night open bus tour on your first night. Look it up online.. definitely worth the money! Gotta go teach, but I'll write more later.