There is a great site on bullseye that also suggests pistols to start with:
http://www.bullseyepistol.com/
But as the others posted, .22 semi-auto is the best way. I thought I new how to shoot "accurately" when I got into bullseye. I was very, very, very wrong. There is a lot to learn, and .22 is the best way to learn it. The good news is the skills carry into other shooting disciplines.
For what its worth, I have been shooting bullseye for about a year, practicing daily on airpistol and dryfire. But I only own a .22 and am only now shopping for an accurized .45. I did not even start borrowing other club members' .45s until a few months ago, I just shot the .22 leg of the match. This was not because of a lack of means, rather I saw that getting a .45 without really knowing what I wanted/needed was pointless.
In my opinion, bullseye will open your mind to aspects of shooting you didn't know existed. Particularly the importance of a gun that fits you anatomically, and who knows if at this stage you even know how to grip the gun correctly for bullseye (I didn't). There is a lot that goes into getting a 1911 that fits you right.
I could go on, but my point is there is something to be said for getting a .22 like a Ruger Mark II/III or S&W 41, then wait a few months before even thinking about a 1911 .45. The one exception is if you were positive you wanted to shoot a Marvel conversion on top of a 1911 lower.
Of course, the question becomes well what do you shoot in the centerfire and .45 stages? Easy, either borrow, or bow out of the scorecards and just shoot your .22 for practice in the next 2 stages.
I've never seen it, but the general advice is that moving to .45 too early in the bullseye shooter's development will ruin the shooter if he/she hasn't yet mastered the .22. If you want a goal, then I'd say 805-820 on the .22 leg is when you can say ok i'm good enough for a .45.
Remember, bullseye is a one-handed match.
Good luck...kep em in the x-ring!