We camphost at Point Lookout, MD each and every summer and we go into DC at least two or three times each year.
We have 2 DD's, aged 11 and 8, and they LOVE it. We drive in to Suitland and his the METRO just after rush-hour, 9:30ish. We have never bought tickets for anything other than getting online and signing up for tickets to the Washington Monument. There is a fee of $1.50 per ticket purchased online and picked up at the WILL CALL window.
You can pick up a METRO map in any station and then sit down somewhere cool and take a really good look at it. Mark the places you want to go and the closest station to it. At many of the tourist destinations (always at the Arlington Cemetary), there are people there with tourist maps and they want you to do the "bus trips", but they will give you the map. This is a great map.
An all day METRO card is $14.50, but if you go to
Fares, there are many reduced fares - disability, 65+ and all sorts of others. You can get a 7-day card for less than $60 per person. There are all sorts of benefits.
You can also print out of METRO map if you have access to a printer.