The tea shops in the DC are are not great but there are a couple worth stopping by if you're looking for something to do. Since they also serve food you might want to stop by at lunchtime. First, in Georgetown is Ching Ching Cha. They specialize in Chinese teas. The owners are from Hong Kong I believe. It is a nice setting to have a pot of tea and the lunch meals are pretty good. The tea they sell is packaged in plastic bags that I believe let the tea go stale quickly though. They have a decent selection of tea gear but nothing exceptional.
There is also local chain called "Teaism". There are 3 DC locations. The Penn Quarter location has a restaurant as well as a separate shop next door. The DuPont Circle location is in a fun neighborhood. The tea and teawares are ok. I have only been there a couple of times. The food is decent.