We're going in July - I grew up in / around the area but my kids have never been, so we are doing full on tourist mode.
My kids are 10 / 13, so we are not doing a lot of the war memorials this time. We are, however, working on a Council's Own Girl Scout badge for my daughter, which is driving some of the things on our list below.
I would not bother with a car - the DC public transport system is easy, reliable, and relatively safe (be smart......) The caveat would be if you plan to take a day trip or two, say to Mt Vernon or Baltimore. You can buy prepaid metro passes online and you can take the metro straight from the airport if your hotel is on the metro line.
I prefer residential hotels so you can eat breakfast / snacks at your leisure, and possibly other meals, depending on your schedule. Consider using something like a grocery delivery service or Amazon (Prime Pantry!) and having stuff sent to the hotel.
If you want to tour the White House, Capitol Building, or go up the Washington Monument, get on those now, as you will need to book further in advance than you'd think. If you are doing other 'pay to play' items, go ahead and research ticket availability and prices.
These are the things currently on our list - we still have a pending request on a White House tour but have been warned those are hard to get approval for, so we are planning as if we won't get it.
Washington Monument (no elevator, missed the availability window)
Jefferson Memorial
FDR Memorial
Lincoln Memorial
Boy Scout Memorial
Fords Theater (not free)
American History Museum (free)
Natural History Museum (free)
National Archives (free)
National Art Gallery (free)
Air & Space Museum (free)
Constitution Hall / Daughters of the American Revolution Museum (free)
International Spy Museum (not free)
Capitol Building tour (free but requires a reservation for security purposes)
Supreme Court Building tour (free)
White House (external photo op only)
National Zoo (free)