Parking is $22, and if you plan to park you need to arrive early, well before 10:00am. With a FONZ (Friends of the National Zoo) membership, you get FREE Parking for one year, plus other discounts and invitations to special events. There are Metro stops just north and south of the zoo, but we've never tried that: 1) because I didn't want to try it with a double stroller and loose kid and 2) because now that they're older, it costs less for all of us to park than to ride the Metro.
If you're bringing small children or a stroller, I recommend that you park in the lot at the bottom of the hill. You'll enter the zoo in the Kids Farm area, which includes the Pizza Playground, then head up the hill while everyone's energy is high. Then, when it's time to go, it's a downhill walk to the car. We generally hit half the zoo in any one trip, no matter how old the kids (or adults) are that we're with. If you're planning to do the upper hill areas (elephants, Asia Trail/pandas, and cheetahs), park in one of the upper lots if you can find room. The paths around the zoo are quite steep.
Every part of the zoo is a hit with my family, especially Amazonia, an enclosed rainforest. We also love the Small Mammal House (there's some weird creatures in there!). Last, we usually spend over an hour in the Great Ape House, with the gorillas and orangutans. My kids love to sit and watch them interact, especially when there's a baby running around making trouble for the older ones.
And be sure and follow my Mid-Atlantic Family Trips board on Pinterest.