Thanks for all the good info, everyone. After reading all of this and talking to a couple people, I think an AR is in my future. Now to figure out optics, barrel twist, and other accessories.
I wrote this a couple of days ago, but didn't get a chance to post it until today. Some other people have already posted excellent info, so some of this may be unnecessary---but I already typed it, darn it, so I'm posting it!
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There are certainly people here who know the ins and outs of rifle builds much more than I do, so I'll leave that up to their expertise---but I can at least talk a little bit about 3-gun competitions, and what works for it.
Couple of assumptions:
1) You mean modern action 3-Gun/Multigun competitions, not Cowboy Action competitions,
2) You plan on trying a number of local matches and working on that before you jump into larger matches, and
3) You don't really have any flavor of AK or AR at the moment, and are looking for something to get yourself started.
Under those assumptions, here's a couple of thoughts:
1) AKs are good guns. That being said, almost no one uses them seriously for competitions past the local level. (About the only time it happens is if someone has lost a bet…) Reloading them isn't nearly as fast, and it isn't as easy to be accurate with them---and there aren't as many aftermarket parts available for them to customize them to exactly what you need. So---get an AR-15.
2) In general, until you've done some competitions, you don't really know what you are going to need---and if you look online, you are going to get so many differing opinions that you'll go nuts trying to figure out which one is "best".
Since "best" is highly dependent on you and what type of 3-gun competitions you attend (and where), chances are if you take an "expert's" word for what you "need," you'll spend a lot of money on something that you are going to significantly change over time anyway.
So---get yourself a basic, reliable AR. Doesn't have to be high-end, doesn't have to have a ton of aftermarket parts and competition-level gadgets on it. Yes, you can do perfectly well with an entry-level DPMS or Bushmaster, to start. If you want to spend a bit more for a better-quality AR, go ahead---but don't worry about all the bells and whistles yet. After all, you don't know what you are going to need yet.
3) What division will you be shooting? For example, are you the Shotgun King, and already have a Saiga setup? If so, you are going to be in Open division, which means your AR better A) have a GOOD quality scope on it, set at a nice high magnification, and B) better either have angled irons sights on it, or a 1x red dot for close targets. (Preferably a red dot. And your pistol better be Open-division level, or you are at a serious disadvantage.)
Or since you are just getting started, are you more of a regular person, who is going to shoot Tac Irons or Tac Optics? Which? If Irons, then just buy an AR, and you are good to go, locally. Tac Optics---then you'll need a decent scope for your rifle. Some people just put a red dot on it, some a 1-4x scope, some have a better scope with angled iron sights---any of the above work. But if you put any scope on it at all, you are in Tac Optics, so make sure that whatever you use, you can shoot (accurately!) at targets from 5 yards to 100 yards (for local matches) or up to 500 yards (!) for large matches. Our last local Multigun match had 10" plates at 100 yards, and full-size paper silhouette targets at 3 yards. A good 1-4x red dot is great for that.
So...
Some people say build your own, which is perfectly fine. (Obviously that depends on how much knowledge you have already.) On the other hand, there is nothing at all wrong with buying a basic AR and competing in Tac Irons division for awhile, to get some experience AFTER which you can figure out what you need.
There are a ton of good, solid, basic AR-15s out there at excellent prices. Any one of them will be perfectly acceptable at local matches in Tac Irons.