For an entry level gun that you can shoot right now, I'd go for a Baer. You can find them on the shelves on occasion.
As one poster mentioned already, check out the Bullseye List. It is an e-mail list, you can subscribe easily by going to
www.bullseyepistol.com.
One poster mentioned HK. First of all, I love HK handguns; one of the worst moves I've ever made was trading off a USP40.
That being said, even though an HK is combat accurate, it is not bullseye/conventional pistol accurate.
A good bullseye pistol (wadcutter or ball) will consistently deliver 1 1/2 inch to 2 inch groups at 50 yards. Yes, I meant YARDS, not feet, and these are minimum five round groups. The minimum standard acceptable is a 3 inch group at 50 yards. Check out the articles at the site mentioned above for a detailed explanation into why you want a pistol that shoots this well.
At the same time, prepare to make an investment into your pistol. You will in all probability want a dot sight for your wadcutter gun; the best on the market that is affordable is the Ultra Dot.
To get the most from your bullseye pistol, there is a good chance that you will want a custom grip. There are some excellent options; for semi custom grips that will help, take a look at Vitarbo grips, or some of the excellent offerings of Karl Nill (Nill grips).
By far, the best (IMHO) custom grip manufacturer is Randall Fung. Take a look at his website. He makes true custom grips from scratch, built to fit you.
Do you reload? Prepare to start if you don't; unless you have a rather liquid source of disposable cash, centerfire match handgun ammo is EXPENSIVE. The standard for wad guns is Federal GM Match, 185 grain SWC ammo--and it runs about $40.00 per box! Casting your own bullets will save even more money.
This is just the tip of the iceberg--you also need a good accurate and dependable .22 pistol, at a minimum. If you want to go Distinguished, you'll need a dedicated ball gun too--plus there are other items that will make your life a lot easier on the firing line.
My personal bullseye kit consists of a .22 with one backup gun (Hammerli 280, backup is a Marvel conversion on a 1911 frame), a SW M52-1 for centerfire, and a custom 1911 wadcutter gun built by Ed Masaki for the .45 stage. My hardball gun was also built by Ed Masaki, and also serves as a backup gun for the wadcutter gun. I cast my bullets from pure range scrap
; I use the 200 gr. HG 68 LSWC, sized to .452, on top of 3.9 of Clays for my wadcutter gun. It saves a LOT of money.
The most important part of prep for a BE match is practice--and lots of it! Shoot, shoot, and shoot some more.
Best of luck in your quest. Bullseye is, to me, the most challenging type of handgun competition--and I enjoy it immensely! See you on the firing line!