Good call on thinking of a dirtbike as a learner. There's really a lot of stuff you "learn" on one of those that can help you out on the street. And by "learn" i mean crash, pick up the bike, odds are undamaged, kick start the sucker, hop on and keep riding, and think to yourself "man, the next time i hit a rock like that, i'm NOT going to panic and lock up the brakes". There's more to it than that though. For starters, the unpaved ground is only as hard as the largest rock. Try to avoid those when crashing, and you won't be in quite as much pain as crashing on pavement. Also, rash isn't as bad sliding on dirt as it is on pavement. BUT that being said, going from a XR250R (about as big as i would recomend for a new dirt rider, and NOT a 2 stroke) to a GSX-R 600 isn't the smartest of things. Yes, the XR teaches you how to deal with your back end kicking out at inopertune times, and yes, it does teach you good recovery from obstacles, but the power it puts out is no where as extreme as the GSXR. Sure, it weighs half as much, but the GSXR has well more than twice the power. Also, all that dirt riding doesn't preoare you for dodging cars as much as one would think. Now, if you're set on riding a dirt bike and upgrading to the 600, go for a dual sport. Something you can blast down a back woods trail, then ride the superslab to get home. I, myself am a big fan of Honda's XR650L. Always was, and always will be a big fan of Honda's XR line ofhdirtbikes, even if they're primarily making CR's and CRF's nowadays. There's also the kawasaki route, the KLR650. Not as offroad capable as the honda dual sport, but a lot better on road than the honda. Better range too.
Just be aware, these dual-sports are only OK at what they do. They're OK road bikes, and OK offroad bikes. So don't expect to break any land speed records, or hit 50 ft jumps with them.
Now, some more of my opinion

. You can still get a 600+ cc sport bike as a beginer. The Ninja 650R and SV650/S are good beginer bikes that will keep you happy for a long time. And not only can they perform, but they're friendly enough that a little too much throttle, or a bad launch won't send the bike into a 12 o'clock wheelie. Sure, they WILL wheelie, but any bike can wheelie. My own experiance has shown me that you don't NEED a full on sport bike to ride fast. I've had to stop and wait for my friend on his YZF750R, and actually had my friend on his 929RR pushing his limits to catch me (though we both nearly "poo poo" ourselves on those particuler corners, nasty corkscrew followd by a tight left hander, than a decreasing radius, blind, right hander with a pavement transition at the apex)