The DC metro is one of the best public transit systems in the world. To my mind, it's part of the DC experience. As with any rail system it's a nuisance from time to time but trying to drive downtown will not be a breeze, either. Besides the traffic, there are lots of restricted lanes and other oddities on the highways around DC that can throw you for a loop if you're not used to driving there. At least get a good map beforehand and figure out your route before you get under way.
I would skip the Nat'l Zoo. Unless it's been massively upgraded in the last ten years, it's small, drab, and pitiful. It's really just there b/c the nation's capital needed a zoo so they squeezed one in where they could. IMO, Old Town Alexandria does not merit a stop if you're only there for a few days. Great restaurants, but that's about it. If you want a break from downtown, go to Mt. Vernon.