Seconded. Do what this man says. UJM or dual sport is the way to start, especially dual sport. I wish I started on one. Height may be an issue for you on that though.
If you absolutely must have a sport bike, get a Ninja 250 and don't pay more than $1500 for it. Year is unimportant. Brand new they MSRP at $3000 and there were factory rebates last year, so don't let anyone scam you.
There's going to be a bunch of naysayer friends of yours who will rag on you for having a small bike, but I'm willing to bet that 9/10ths of them don't have a bike, or let their dad pay for the one they do have. I started at age 19 on a '72 CB500. I paid $400 for it, and fell over on it at least 4 times. Trust me, it will be worth it. Whatever you buy, you'll be able to resell for the same price and then you'll be comfortable moving on to a bigger bike. I promise you that you will thank me if you follow this advice.
I'm just a bit taller than you are, so I know the feeling when over half the bikes you can't even ride. My suggestion is a Ninja 650R. Cheap, good motor, enough power to still get you in trouble and you can always replace the bars with clip-ons if you want a more aggressive riding position later.
No brainer: Ninja 250. $3000 new, your feet will be flat on the ground when stopped, insurance is cheap, sportbike look (full fairing), lots of fun to ride. Have put almost 8000 miles on mine since September. By the way: I have been riding since 1980 and have had FZR 1000, R6, SV650-love my little Ninja, and when you learn to really ride, it romps through through the twisties. GO for it!
I am in Burbank California and it is a pristine 1970 CL350 in tangerene orange with 8000k on the clock. Looks/runs like new. I will let it go to a fellow MOron for real cheap, like $900.00 bucks. Injury forces sale.