Summer camping in DC with kids.... Haven't tried it, nor would I, but there are some campground / state / federal parks within ez reach via train / then metro.
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How old & how many are the kids? When ours reached ~ age 12, they were on their own in DC. We met at lunch (basement of U.S. Customs Building; Food Court), shared ideas and spent the afternoon together looking at the highlights. Cell phones would make this much ez'r...
Mint is good, as well as the Holocaust Museum, and FBI / Ford Theater (very near each other). Take the Architect tour of the 'Pensioner's Building' (Near police officer memorial). We do Arlington Cemetery a couple times as well as the Monuments and national Cathedral. (evening / morning to beat the heat) 4th of July was grand from Lincoln Memorial, but next time I'm bringing a floating lounge chair for reflection pools.
I find that as a family the transit fees can be burdensome. Thus we have always driven in and had no problem. (except for 4th, we took transit)
For lodging, we started using private guest home directories about 20 yrs ago, and they made our trips very special (and cheap). Basically $10 - $20 / night (Cheaper than campground). + you get first hand info of area, often free transportation and food. (and other kids to play with !)
We started with a homeschool guest home directory, and now use a 'retiree'
(age 50) directory as well as previous ones. Can't recommend them highly enough for a quality vacation experience. I have sent several friends w/families to DC with my directories and they have come back with glowing reports of great fun and accommodations.
One July we took a DC 'break' and went to Ocean City, MD for a couple days. We bought an 'all-day-pass' to a go-cart track on south end. It had ~ 10 different tracks and we joined another family from PNW and had a blast. (kids were teenagers at the time).