First determine the class you want to shoot and the rules you must shoot under.
In recent years, when rifle weight is restricted to some degree, the 6.5 has been dominant. The cartridge most often used is the 6.5-284. Over the years it has set some very amazing results however, quite hard on barrels.
There are many smaller cases to help reduce that problem. 6.5X55, 260 and the AI versions of both are very common now. My 6.5 Mystic is my idea of a better mouse trap.
The goal is to get a 140gr class VLD match bullet to at least 2750fps (2900 to 3000fps prefered). Ballistics are superb by way of wind drift reduction and relatively low recoil.
Over the last 5 or so years, the 6mm has really come into its own. First the 6BR but that had to be run at pretty elevated pressures. Following are a variety of improved versions. Most are excellent but also run pretty high pressures.
To help reduce any risk of locking up actions, some went to the next larger case, the 22/250. Many versions now all with positive results.
David Tubbs has been promoting his proprietory cartridge the 6XC. Of course, it has had a great match record but then David knows a thing or two about shooting.
If rifle weight is not restricted, the 7mm, 30 and 338 offer better ballistics but recoil and shooting cost goes up dramatically.
The 308/223 fall into service/tactical rifle classes and do well when paired against themselves. Not very competitive against the 6mm and larger cals due solely to increased wind drift. Now there are some very skilled Palma shooters that can shoot possibles in a tornado but for the average joe learning, having as much as 50% less wind drift helps.
The 6.5 in a 'better' cal based on paper ballistics but look at match scores and the 6mm is not giving up anything much. New bullets will push the 6mm into the lead.
www.6mmbr.com is a wonderful site to learn all about this stuff. There is no one perfect combo in the LR game, unlike SR BR.
Just look at how many shooters are shooting what classes. Sometimes just having others to shoot with will determine what you use.
For me, shooting in a light rifle class, I would use my 6.5 Mystic of course but the 6mm is getting my interest as new bullets come out. If someone makes a quality 6mm bullet with a BC over 0.6, this cal will take over all light rifle LR shooting.
For the heavy classes, the 7mm would get my nod.
Good luck...
Jerry