Not liking urban riding limits you a bit, but there are a lot of bike routes in the city that are fairly low traffic, especially on weekends.
Here’s a link to the DC bike map.
http://dc.gov/DC/DDOT/Publication%20...11_Fulldoc.pdf

On weekends most of Beach Drive in DC is closed to cars. Good hills climbing out of the Rock Creek valley. Continue on Beach Drive north into Montgomery County and you can get to some more good riding areas.
West on MacArthur Blvd gets you to River Road and some more rural areas towards Poolesville.
From Greenbelt you can get to the USDA Research Facility, Bowie, Waldorf and rural parts of Prince George county..
Some MUPs don’t have much of a crowd. The Anacostia Trail system is an underused gem, as is the upper part of the Rock Creek Trail.
To avoid some of the city riding, you can take the Metro to a station at the end of a line (Vienna, Greenbelt, Shady Grove) and start your ride from there.

Since you’ll be on Capitol Hill, you and your family might consider Capitol Bikeshare for getting around town and site seeing.
And I strongly recommend doing a monuments-by-moonlight ride.

Have fun.