Any of the CMP/DCM models made by the various AR manufacturers would suit the bill just fine. Call the club--most have M1's and/or AR's that they can loan out. I'd recommend shooting ASAP and not buying gear/rifles first--it's easier to know what you'll need and want after you've got a couple matches under your belt. I know at my club, all you need is about $30 to cover the fee and ammo (bring a sweatshirt regardless of the weather).
Most matches are run under NRA rules which allow for Service Rifle (must have the same exterior profile as a GI rifle) and Match Rifle (any centerfire with iron sights that's not a Service Rifle). Even if the club is "a CMP club", they generally use the NRA rules or a relaxed version. The fact that the club is affiliated with the CMP is nice--when you get a membership, you fulfill one of the requirements needed to buy rifles and stuff from the CMP.
The NRA and the CMP definition of "Service Rifle" is pretty much the exact same (infact NRA rule 3.1.3 says if the CMP calls it a Service Rifle, then it's good to go). The majority of guys shoot Service Rifles. The match rifle route is pretty much wide open--Winchesters, Remingtons, Tikka, tricked out AR's (spaceguns), etc.
The rules (section 3 covers the various rifles)
http://www.tngbbs.com/rifle/nra/hp/
The only time the CMP rulebook comes into play is at sanctioned JCG matches and CMP Excellence-in-Competition "Leg" matches (a 50 shot course, no sighters, usually held at the larger championship events). Your typical state championship would be an 88rd championship match fired on Saturday under NRA rules with Service Rifle and Match Rifle categories and then on Sunday the Service Rifle guys shoot the CMP Leg match.