NRA Conventional Pistol (bullseye) is shot in three divisions:
Smallbore = .22 Gotta have it to shoot the whole match, not just a cheap plinker for "practice" whateverthehell you think that is. The rulebook doesn't care about the California "Gun Culture", the NRA has been doing this stuff for over a hundred years and they like stablility. Can be a conversion on the .45 auto but one good enough for serious competition is not itself cheap. Marvel Unit 1 is the only one still in production worth considering.
Centerfore = .32 or larger. Used to be a lot of .38 Special revolvers, then .38 Special autos, but most folks just use the same gun as for...
Bigbore = .45 ACP Period. Mostly accurized 1911s but there are a very few others like S&W 945 and Pardini.
Now, you are not going to get all that done for $600. You are not going to do it with a .380.
I suggest you find a club to shoot at, find out what events they shoot, and see what it will take to get equipped. You might even get personally acquainted and buy secondhand guns and other gear from somebody moving up in quality or quitting due to poor health.
You could shoot smallbore only, a .22 is a Real Gun and a good one is fantastically accurate on medium priced ammo.
You could shoot centerfire only, S&W has put the K-38 back into production. I don't know the price but it will likely be less than an auto anywhere near as accurate.
You could shoot bigbore and centerfire with the same gun. Get a 1911 that is on the Schwartzenegger Reich Approved Handgun Roster and shoot it. When you are good enough to need an accurized gun, I believe you are allowed to have an existing gun worked on.
As a college student, you are no doubt accustomed to studying stuff you see no immediate use for. NRA rules are online at:
http://www.nrahq.org/compete/nra-rule-books.asp