Georgetown is a fun place to walk around ... stores and restaurants, but there is no metro there to travel elsewhere. You'll need a car. Many hotels have parking (like the Holiday Inn Georgetown which is a little removed from the main part of Georgetown though), but you sometimes have to pay a nightly fee. I went to school there and have been back many times to visit. I've stayed at the Mandarin Oriental which is right next to the mall (where all the museums and monuments are), but it is VERY VERY expensive. If you stay in Woodley Park on Connecticut Avenue, you are right next to the zoo (which is free) and the Metro is also right there. You will have to ride the Metro (subway) to get down to where the museums are. Dupont Circle also has lots of shops and restaurants and also a metro stop.
NW DC is the nicest of the four sections. Of course, if you are right on the mall (where the four quadrants meet) the NW, NE, SW, SE doesn't really matter.
The metro is easy to figure out and safe to ride. It's the cleanest subway I've been on.
If you have a car, take a ride to Old Town Alexandria. It has beautiful cobble stone streets, a marina, shops, restaurants, and gas street lamps. Very pretty. Also if you have a car, a great place to get brunch is the Carlisle in Virginia right outside DC. They have really, really good food.
Make sure you see main museums (Air & Space, American History, Natural History). The Holocaust Museum is incredible, it's presented so well, it is very, very, very moving. The WWII memorial is relatively new and very nice and don't forget the FDR memorial which is often overlooked, but one of my favorites.
I usually stay with friends when I go, but when we want to play tourist we will sometimes go and stay in a hotel. Like I said we stayed at the Mandarin once which was incredible, but a one time thing because of the price. Generally, we stay in Georgetown, but we drive to DC so having a car helps. If you can find an affordable hotel right on the mall, stay there. Otherwise, try those other areas I suggested on the Metro line if you don't plan on renting a car.
Have fun! If you have any questions, I'd be glad to try to answer them. PM me.