First, welcome to DRN :)
There are several mfg's that still make 2-strokes, of course the new ones will be at leat 2.5x your budget. Fortunately, there are a lot of older used bikes out there. Based on the info you provided and lack of experience on MX'ers, I'd suggest finding a clean 125. They are a BLAST to ride and will teach you a lot for when/if the time comes to move up to a 250.
Have you got anyone one that's knowledgeable about bikes? They are certainly easy to work on, but maybe someone could help when you are out looking for a used one. Having a mechanic look one over before purchase would be huge. In most cases you can expect some up front costs... a top end and smaller stuff like fork seals, cables, tires, all that stuff adds up so allow some room in your budget. Keep in mind, plastic (body parts) are cheap, focus more on the condition of the rest of the bike. You want to buy from an owner that can tell you more about the maintenance he performed that the bolt on bling he added. The closer to stock the bike is the better.
Keep in mind, if you go with an older bike, parts may be hard to come by. When you find one that you're interested in, go to a site like
MotoSport.com: #1 in Dirt Bike, Motorcycle, ATV and UTV Parts, Accessories and Gear | MotoSport , and plug the bike info in to see what all they have available.
I'd look for anything 2004 and newer. I happen to have seen a really nice 2006 Yamaha YZ125 recently that went within your budget.
As far as brands go, buy the cleanest, best maintained bike and don't worry about what color it is. A new rider is a LONG way from noticing any differences in brands, model years. DO NOT be tempted by the off brands. Stay with the big names, Honda, Kawasaki, Suzuki, Yamaha, KTM, etc. You'll also want to stay away from the trail bikes each brand offers.
Good luck! And again, welcome to DirtRider.