My wife is going through this very process. She borrowed my Surly Cross Check (cross bike with road tires and drop bars) last year to do a sprint triathlon. It worked well for her as a newbie into the sport, though the frame was too big for her.
This winter, I bought her a Trek FX, which is essentially a road bike (skinny tires) with a flat handle bar. She can use it for a triathalon and for trail riding as well. If she sticks with the triathalon, then I could see her buying a good used road bike for $300-500.
To summarize, I would suggest either finding a used true road bike for $200 that would let you compete for awhile, then if you decide you want to stick with it, you can upgrade to something newer/nicer.