Chances are you'll be feeling the heat in August, so head to the beach! The drive from DC to  Rehoboth Beach is gorgeous, and you can take some time to stop at  Annapolis for even more.  The beach is an experience (read: lots of people!), but the drive is full of lush, greenery and more "country" experience than I anticipated on the East Coast.  On your way back, stop for a crab dinner at  Harris Crab House in North Grasonville, MD!

Definitely try the United States Botanic Garden, any of the museums are free, and even a ride on the subway will take you to the United States Naval Academy.

There is a ton to do in the areas just surrounding DC, so I'll give you my favorites. 

For nature within the city, go to Rock Creek Park.  You can bike it, walk it, scooter it... etc.  You can also walk along the C & O canal near  Georgetown (Washington DC) and take a boat ride pulled by mules.

Just 30 minutes north of the city, go to Great Falls National Park.  You can see the falls and hike around the area on either the Virginia or Maryland sides.  The Virginia side is closer to DC.  If you go farther north, you can get to Harpers Ferry.  The city itself is really incredible and is also part of a national park.  The area is surrounded by water, so check out one of the many places in the area that offer river rafting and ziplining.

South of the city by an hour, go to Flag Ponds Park in Lusby.  It's the nicest beach in Calvert County, Maryland.  The water is warm, the beach expansive (and not crowded), and there are tons of shells and fossils including shark teeth from the Miocene age.  There's also a fishing pier, so bring chicken necks and catch the infamous blue crab like the locals do.  If you get down here, you may want to check out the Calvert Marine Museum in Solomons.

Annapolis itself is gorgeous and worth a visit.  The best way to see the city is by boat, so rent a boat and enjoy.  We've rented the Duffy Boats and have enjoyed it.  It's simple, but comfortable, has space for a picnic, and a bathroom (great for kids). 

Easy: Great Falls National Park - specifically, the Billy Goat Trail(s). Also: Mount Vernon Trail

Great answers and suggestions so far.  Totally agree with every one that Tiffany mentioned.  I would add the United States National Arboretum to the list.  It is free and a beautfiul place to spend a couple hours.  August will be hot and humid so best to do this one early in the day.  I actually wrote a post about my solo visit there on my blog at http://www.ttstrippin.com/1/post/2013/05/a-spring-day-at-the-us-national-arboretum.html.  You could easily do the Arboretum in the morning, then head to Annapolis in the afternoon and follow it up with a drive across the Bridge and over to Kent Island for a scenic drive.  I'm not sure where you are coming from and if you have been to any of these areas already.   Great Falls National Park should definitely be at the top of your list.  You can use the DC circulator ($1) to go to Georgetown quite easily (I would not recommend driving there--traffic and parking too stressful/busy).  Definitely use the public transportation for all things within DC-between the metro and the circulator, you can get everywhere very cheap.  No need to buy the expensive hop on/hop off tickets.