As others have mentioned, you could be there for weeks and not see everything. Make a list of everything you'd LIKE to see, then pick a few per day depending on geography and on your stamina. Try to vary the type of activity, so you don't get "Museum'ed out". Also, don't do too many sad things in one day such as memorials and the Holocaust museum. Keep an eye on museum hours; they close as early as 5:30. The Newseum is worth paying for (and a ticket is good for two days); the Spy museum, not so much unless you're a fan of that stuff and young at heart.
That's a good tip about getting off the Metro at Cleveland Park so you can walk downhill to the Zoo entrance, then when you leave the Zoo turn left when you exit so you can walk downhill to the Woodley Park station. The animals are out and active long before the buildings open; go at 8 AM if you want to see pandas frolicking.
I like the off-the-beaten-path places like the Building Museum and the Folger Shakespeare Library, with its Elizabethan theater.