I would go to a local type Fly Shop, look them up on google etc. Frequently they have instruction for free. If you can't do that, I would purchase a very basic video online and study that. I found a video pretty helpful in getting started. It is a study of technique, not strength. No one in my family was ever into it but under the right conditions it is loads of fun. It does have limitations depending on where you are going to fish, etc. I would probably also go to the local library and see if they have any beginers books on fly casting. DON'T just go out and buy a fly rod. I have had three, and like a jerk, I over spent on my first one, and became frustrated. I now have a decent Sage rod, built for more the beginer caster, and it is much easier to throw. I have also found that if I go up on weight on the fly line, it is easier to throw. If I have a 7wt. rod, I use 8 wt. line. Good luck.
I just bumped an old thread "Roll Call for Mid-Atlantic Non-Chain FLy shops". Thats a pretty complete list.
For your area, you have a few choices:
Angler's in Annapolis has a small fly section, with a large assortment of fly rods. They used to be an Orvis dealer, but the sign disappeared off the front recently. I think their fly fishing person is only there a few times a week.
Angler's Lie in Arlington. They are a full service fly shop with a complete assortment of rods, reels, lines, tying materials, books, clothing, etc. They're a great shop, but they're a little inconvenient for me to get to. If you work in DC, that might be another story.
Orvis in Arlington. Jason, the fishing manager, just recently joined the board (rofa). Its an Orvis store, so they carry pretty much anything thats in the Orvis catalog.
As for casting lessons, Joe Cap is a great guy to go to. There's a recent thread with his accolades. Someone can chime in with his contact info, or you can call WCO for it.
WInchester Creek Outfitters in Grasonville, MD on the Eastern Shore. They're a full service fly shop that most of us from this area go to. They have a good selection of gear, materials, etc. And can usually order anything specific. They also have a large casting lawn behind the shop for trying any of the rods.
I recommend Winchester Creek, but I will also go to other shops if I'm in the area or am looking for a brand that WCO doesn't carry. I like to spread the wealth.
The only request I do make is to support your local fly shops. You'll get better customer service than the big box stores and more personal attention.