The first thing I would do is find out what is going on locally. Having local people to shoot with makes the game much more fun, not to mention the knowledge that can be attained from many of the experienced shooters.
Second would be to find what interests you. There are competitions from 100 yards out to 1000 and anywhere in between. Certain calibers and chamberings do better for different distances. Before you can pick a rifle/chambering, you need to know what distance you plan to shoot. It is also important to pick a distance you can practice at. If the local range only goes to 100 yards, practicing for further competitions is going to be difficult.
Now that you have found a place to practice and know what distance(s) you want to shoot, you can pick a rifle that fits that bill. You also need to look at cost. The bench rest game can be inexpensive to start into, but to be competitive, it gets expensive very quick.
Rimfire is certainly the inexpensive way into the sport. There are many used dedicated .22lr benchrest rifles out there that will let you compete for at or under $1000. Add on a Weaver 36x scope for another $300-$400, and $500 worth of a rest setup and a couple hundred in wind flags and you could be in the game and have equipment capable of winning some matches for right around $2000.
Now you can certainly do it cheaper. The Kimber example above is a good one. Buy a $400 Kimber 82G, sell the sights for $150 if you want (though I would keep them, very nice sights), buy a Nitrex(made by weaver in Japan) TR1 6.5-20x scope(clearanced by natchez plus they have a $50 factory rebate) , a Caldwell Rock BR front rest, a rear bag, and you are in the game for less than $1000. It won't be world class competitive but it may work just fine at the local level. The good thing here is that the Kimber is under priced at $400 and you can easily get your money back. Same thing with the used BR rifle, it has already taken the bulk of the depreciation and you can get most or all of your money back out of it if you decide to sell.
Rimfire is good for a few reasons. First, you have to know how to read the wind to be competitive. This only makes stepping into the centerfire world that much easier. Second, the equipment lasts much longer. A rimfire barrel can go through 10s of thousands of rounds and still shoot well. A centerfire barrel needs replaced after a couple thousand. Also, rimfire ammo is cheap. Top tier ammo runs about $15/50 rounds though if you buy in bulk you save about 10% from that or so. That puts the cost at about $0.22 a shot. You can't get a premium centerfire BR bullet at that cost, let alone loaded ammo. Even handloads run into $0.50 a shot on the low end and range up from there. If you step down a notch on ammo cost as well into the mid range ($4-$6/50) price range these savings grow even more. A 5000 round case of Wolf MT ammo runs about $400 ($0.08/shot). 5000 rounds of centerfire will cost you at least $2000 in ammo plus a $500+ rebarrel or two. As you can see, it quickly is more affordable to be in rimfire BR shooting.
That said, there are a lot of centerfire competitions out there shooting from fairly close to way the heck out there. I know the local guys for me shoot from a bench but they aren't using world class BR equipment. For me, 1/2 MOA out to 200 yards is what it takes to be competitive locally. That can be done with the right factory rifle these days. This certainly isn't competitive BR, but its all I enter locally in the BR world so it works well for me. Check out benchrest.com and 6mmbr.com in the classified section. Lots of good deals can be found on there for some used centerfire equipment ranging from accurate factory rifles up to full blown custom dedicated BR rifles. You can see the pricing and buy what fits your desires. Again though, realize that barrels are like bullets in that they are consumables. If you want to get competitive, you are going to go through a barrel a year or more in the centerfire game.
