Our Cub Scout Pack just did a family style trip. Everyone was on their own for transport and lodging but we were all together for some sight seeing. Number one is to call your congressional rep or senator. They can set up a tour of the capitol building with not much fuss. They will also arrainge a tour of the White House. Be aware this needs to be requested months in advance but the White House won't let you know till 4 weeks prior.
After those tours all the Smithsonian Museaums are free - www.si.edu
and there's about 12 of them - Air & Space, Natural History, American History, American Indian are all popular. While on the Mall you can arrainge for entry to the Washington Monument www.nps.gov/wamo
Tkts are free if you stand online early in the morning but the number given per person is low. You can reserve ahead for $2 per tkt. Call them to see about Scout Groups. The Bureau of Engraving is also a good bet www.moneyfactory.gov
(not joking on the URL). The 3 war memorials and the Lincoln Monument are all walking distance to the Washington Monument. Don't forget the Scout Monument on the Elipse between the White House and the Washington Monument. The Jefferson and FDR monuments are beautiful but a bit off the mall (by about a mile) and not near a Metro station.
The Metro is a great way to get around but beware, the fare is based on distance and the all day tkts aren't good till 9am. For a guy used to the NYC Subway it took some getting used to.
We didn't try to stay at the Military base. Warnings about potential lockdowns or lockouts due to security caused us to not even ask. I know our Troop stayed at Bolling AFB for next to nothing $ and ate at the commisary like kings for about $3 a head. But that was pre 911 and pre ACLU v DOD so I really have no idea if they'd even allow it any more.
One warning, and since you're from SW PA it might not be a surprise but it can be rather hot, humid, and uncomfortable in WDC in the summer.