Rifle of Choice
If you want to shoot CMP matches (Excellence in Competition matches, the P-100 and NTI at Camp Perry) and NRA matches, an AR-15 in a Service Rifle configuration is pretty much THE rifle to get. Nothing else worthwhile, unless you want to get into a true Match Rifle.
Good rifles are available out of the box from Rock River, Armalite and even DPMS:
Rock River NM
Armalite NM
Also as an option, you can buy an upper from
White Oak and mate it to a lower of your choosing.
Expect to drop between $1000 and $1500 for the rifle.
Really, you need a free-float tube, 1/2-MOA (or finer) sights and a good trigger. Fine points will vary on experience and preference. Slings will vary also; some use the cotton M-1 type, others of us use the 1907-pattern.
Oh yeah, and practice practice practice. Dry-firing at home helps, as does range time at reduced ranges and at full-course ranges. Trigger time helps, but it has to be quality, well practiced and rehearsed trigger time. Blowing ammo for the sake of making noise is counter-productive to good habits. (I use a .22 or pistols for those gotta-blast days.)
Get into reloading, as it will save you a bundle of money on good match ammo and is the ONLY way to get satisfactory 600-yard loads IMO. I started off buying into the "shoot cheap ball ammo (or similar reloads) for practice" philosophy, but quickly switched to shooting my best bullets all the time. Most ball ammo is good only if you're happy to shoot edge to edge on the 7" 10-ring at 200 and 300. Otherwise it is a waste of barrel-life on a match barrel to shoot anything but the good stuff.
Lastly, my personal rifle is an Armalite M-15NM with stock sights, stock trigger, a WOA 1-7" Wilson barrel (replaced the original). It works for me, shoots accurately and is very forgiving to shoot. I use a Turner Biothane All-weather sling of the 1907-pattern. Other gear can be added as you grow, but a good rifle will stay with you.
Good luck and stick with it! A few matches and some serious practice and you'll be surprised how well you can shoot and how far you can do it.