I respectfully disagree with Majic. I shoot Bullseye, and before that I shot International, a similar discipline, and I think that for an entry level shooter, you can buy a great pistol for less than $300.
I personally started on a stock Ruger Mark II Target, and I think those are an excellent choice. The Mark III is the current model. In recent years, these have gone up in price, so on a price/performance basis, I think there are better choices. Although I don't own one, I would recommend the Browning Buckmark. Both of these pistols, in stock form, or perhaps some grips, will easily get you to expert level or higher. At that point, you'll have the experience to know whether you want to put some custom work into your current pistol or upgrade.
If I had to do it all over again, I would get a quality, base model 1911 (with a trigger job, but no other accurization) and a Marvel conversion kit (
http://www.marvelprod.com/). This is more than your $300 limit, and you don't need a .45, but this combo will last you a lifetime. Plus, even if you get out of bullseye, you'll have a quality 1911 and a top notch .22 on the same frame. If you already have a 1911, this is the perfect way to go for you.
Good luck!
-John