I hunted for years with a Rem700 25/06 in both BDL and Sendero, shot several deer over 450 yards and groundhogs at 600+. AMAZING caliber and must be handloaded to be appreciated.
My choice would be .243 (or the before mentioned 25/06), easier on the pocketbook when you reload (which you will eventually do if you want to really appreciate decent groupings), easier on the shoulder, much flatter and easier to learn than a .308. The economical surplus loads are fine in your AR10 or other semi auto .308 but wont quite cut it for target or distance shooting. The real cheap corrosive ammo will erode your barrel WAY faster than erosion in a .243 so that rules out all of the real cheapo ammo.
IMO (and yes I do own a Rem700 .308 Tacticool...) there are many better choices ballistically than the .308 all of which, including the .308 will only be appreciated when reloaded. Again, IMO the popularity of the .308 comes from it being a NATO service round and everyone wants to make a "sniper rifle".
All this said I love my .308, however if I only had to have one gun it would be the .243 (25/06 would be mighty close it just doesnt have the massive bullet selection that the 6mm bullets have, and can be a powder hog). If I lived out west and needed a larger bullet for bigger game then of course this would change....
.223 is a great choice but really isnt very practical if you want to shoot anything over 400 yards (plus all of the limited hunting abilities of the caliber), if you wanted to stick with .223 my vote is to get a nice varmint upper for your AR and improve the trigger.
.243
6mm rem
25/06
.308
30/06
7mm
That would be my caliber order without getting into wildcats.