I'm a bit new to this board, but saw your post and wanted to chip in. I live in Richmond so we go to DC fairly often.
Where to stay?
Cost is obviously a big factor in a large city like DC, and being in the heart of the district typically means $250+/night for a nicer hotel. We've had some good success using Priceline or Hotwire to get 4-star hotels in great locations for a much more reasonable price, like in the $100-150/night range. Takes some research to get in area you want, but I think you can't beat actually staying within a short walk or Metro ride to reach all the sights.
What to do?
When it comes to activities that don't involve eating (we do a lot of that in DC
), the only expense we really have in DC is Metro costs. There are more things to do and see in the main DC area that don't cost any admittance than you'll have time to cover. The boys would probably really enjoy the Smithsonian Air & Space Museum, and depending on their thoughts on museums all the Smithsonians are great. American History is one of my absolute favorites (you get to see the actual Star Spangled Banner!).
Just walking around the National Mall and surrounding area can take up a whole day taking in the sights of the monuments, memorials and historical buildings. Make sure to go see the Korean War Memorial at night, it's awesome! And our favorites building to see the inside of is the Library of Congress — the decorations on the ceiling are quite magnificent.
Other museum-type activities that are good are visiting the National Archives, Holocaust Museum and Bureau of Printing and Engraving to name just a few. There are some tricks to planning ahead for these which you can research some online (or if you care to know more I'm happy to elaborate).
There's a ton more beyond this, but these are some of our favorite go-tos...
Where to eat?
This, too, has endless possibilities, but I'll just share some of our favorites:
Dinner and/or full-service
Founding Farmers - upscale/Southern-ish comfort food
Matchbox - really good artisan-type pizzas
Cava Mezze - Mediterranean / tapas
Casual
Cava Grill - same owners as Cava Mezze, but quicker and casual. Think Chipotle but with Mediterranean food
Good Stuff Eatery - great burgers, fries and shakes, founded by a Top Chef contestant (Spike)
District Taco - great fresh variety of tacos
I've got some good coffee shops as well which I can go in to you if you'd like. Hope this helps!