I live just above DC, in Murl'in.
The obvious thing to do would be to check any one of the Smithsonian museums. They are free, your tax dollars pay for them, so you might as well take advantage of that. The Air & Space Museum is a must see, in my opinion. The Natural History Museum is also a must see, in particular if you want to check out the Hope Diamond. The U.S. Botanical Garden, which is not a part of the Smithsonian, is also free and worth checking out. It is right next to the American Indian Museum (Smithsonian). The National Archives is also free and usually has a cool, changing, main exhibit. I was there a few months ago when the Magna Carta was displayed. I found it interesting that a document that was written some 500 years or so before the Declaration of Independence is in a heckuvalot better condition.
There is the Newseum and Spy Museum, both are around $20 per perscaccon. The Spy Museum is worth it.
You can always find free parking around the National Mall, but you might have to "circle the block" a few times. People come-and-go all day long but not sure if you will find a spot big enough for an SUV and a trailer. You can possibly find a place to park at a Metro, MARC/Amtrak station, but I dunno. Getting a storage space is probably your best bet.
You may or may not be able to park in the Congress/Senate "parking lot" for free on the weekends. I've done it all day long and never received a ticket as well as others but I've also heard it is a no-no without a permit. If you find a spot, keep it. I personally would try to walk/Metro around town as much as possible as the farther you get from the National Mall the more parking becomes a PITA.
Everything else I know about DC has to do with nightlife.
As you may or may not know DC is expensive, even in non-toursity parts.
I have never been to the Holocaust Museum but from what I hear it is worth it...and sad.
The U.S. Marines Museum in Quantico is also worth checking out, in my opinion.
The U.S. Naval Academy in Annapolis is worth checking out if you are over that way.
If you are going up the East Coast you might want to stop at the NSA (about halfway between DC and Baltimore) and check out their Cryptological Museum. It's really cool.
The National Zoo is...I dunno. The only time I have been there was at a time when they first opened for the season. Not much was going on and many of the animals must have still been in hibernation.
The National Aquarium is in Baltimore/Inner Harbor. It is worth checking out. The B&O Railroad Museum in Baltimore is also worth checking out. You can also do a self-guided "The Wire" tour: just drive around east of Johns Hopkins Medical Center. You can check out the Union and Clipper City/Heavy Seas breweries. Or Brewer's Art or Maxx's Tap House.