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For target shooting you just are not going to beat .308 caliber or .224 caliber for target shooting bullet selection. However, with your rifle selection I don't think it will be as easy to get a fast enough barrel twist for the heavier target bullets in .223. That's too bad, because if cost is a concern, .223. brass and bullets are least expensive and easiest to come by. With a 7 to 8 inch twist you can shoot the long 80 grain VLD match bullets. Their ballistic coefficients are up with the .30's you can run in .308, so wind and trajectory are just as good. But I only see the Stevens in a 9" twist. That will limit you.
The .25-06 will wear barrel throats out sooner than the your other choices and in Sierra's bullets, anyway, the only MatchKing just doesn't have high enough ballistic coefficient for great 500 yard performance. None of their .257's do; probably because of the common 10" barrel twist available for that chambering is too slow in that bullet diameter to stabilize long bullets. The .243 has one good high BC MatchKing available at 107 grains, and a 105 in Hornady, but again, barrel twist is the obstacle. You'd need about an 8" twist for best accuracy with the 107's.
The .308's will come with 10" twist, which, in that caliber is more than fast enough twist for the long 180 grain VLD bullets, which are about as heavy as one usually goes with .308. Longer ones take up too much powder space, even though the twist will stabilize them. So, in your shoes I would lean toward the Stevens in .308.
You do want to learn to reload to tame the cost, if you don't already? Not only is it not difficult to learn to do, but even with the most basic equipment the quality of the ammunition you get is generally higher than commercial loads. The .308 bullets, owing to metals market rises, are not the least expensive any more (the .224's are), but you can still do way better than commercial by reloading. At 500 yards either IMR4064 or Reloader 15 powder and the Sierra 175 grain MatchKing are a terrific combination. The Stevens rifle has the Savage action and barrel system, which have an ever increasing accuracy reputation. That's the direction I'd turn in.