Two types of dual sports
The answer to "where to start?" is, like others wrote, is to get a bike and ride.
Safety
In Texas one can get out of the actual motorcycle license driving test (where the state trooper follows you while you ride) by taking the Motorcycle Safety Course, plus you can get a discount on insurance. So, there are some additional reasons to take it.
Bike
I see the world of dual sport bikes as having three "classes":
(1) dirt bikes with head/tail lights (Yamaha WR250, XT250, Honda XR650, CRF230, BMW G450, etc.) a.k.a. "enduros," light weight (more or less), great on dirt, not something you want to ride across the country on (which will be mostly highway anyway).
(2) street bikes with knobbies or the "overlanders" (Kawasaki KLR650, Suzuki Vee-Strom 650/1000, BMW 1200GS, Yamaha Tenere) i.e. bigger, faster on the highway, more comfortable on the highway, can load with lots of stuff.
(3) wanna be dual sports (Ducati Multistrada 1200, Buell Ulysses, etc.).
Of course there are many other bikes and everyone has their opinion, I think buying one depends on the type of riding you think you'll do. If you are going to knock around back roads on a Saturday afternoon and all you need is a bottle of water and a peanut butter and jelly sandwich the first class (enduros) of bikes are great. If you want to ride hard off road and jet back home on the street, then the first class (enduros) are great.
If you want to ride more than a few hours to your destination, which will likely be mostly pavement, and you want to carry your sleeping bag and tent and a few days worth of food, etc., then one of the larger bikes (overlanders) would be good.
I started out with a Kawasaki KLR 650. (Actually, started with a Honda 50 back in 1975 and went up from there.) I rode the KLR into Mexico from Texas a few times, rode it around local dirt roads, etc. locally, but found it rough commuting to work (45 miles one way). I "upgraded" to a Suzuki Vee-Strom 1000 (too large as a starter bike!), then downgraded to a Suzuki Vee-Strom 650. I commute on the 650 and ride twin track out in the Texas Big Bend region every year (8 hours of pavement to get there). And, I carry all my camping gear and fuel for a week. That's how I got started anyway.
This might work for you too. Mid-size street/dirt worthy bike like a KLR650 or BMW Dakar 650 (single cylinder "thumper") to start, then up from there or stick with that type of bike. A buddy of mine rode a KLR from San Antonio to the Darian Gap, then two years later rode the same KLR from SA to Tierra del Fuego. Another muddy of mine rode a BMW Dakar 650 around the world (7 years)(no prior bike experience). So, the big ***, heavy bikes aren't needed unless that's your gig. I ride on the street more than the dirt, so I like the twin-v Vee-Strom.
Whatever you do. Get a used bike like others have suggested and start riding. Ride on dirt, ride on dirt/gravel/mud roads, ride on the street. Ride it all. But just ride. When you're ready. Pack it up with some bags (hard or soft [everyone has an opinion on these too]) and head out for an overnight, then two nights out, then cross the state line, then.......
It's like Nike, man...Just Do It.
(Oh, good gear is essential. Be waterproof and warm in the winter.)
And check out Horizons Unlimited...they're coming to you!
http://www.horizonsunlimited.com/meetings/USAEast2011.php
Great way to check it all out before you commit.