Either the .243 or the .308 are good choices from an economic point of view. The cheaper the ammo the more you can shoot, the more you shoot the faster you improve.
I just read an interesting article in Precision Shooting where the author chose a factory Howa Varmint rifle in .243 Winchester for competition. We don't hear much about Howa here, and that might be worth checking out.
I thought Nightforce scopes would be the top choice for precision work, but in the article from PS, the author stated that the paper work included with his new Nightforce scope recommends against the use of dovetail mounts, or mounts with built in windage adjustments. I would not use such mounts on a precision rifle, however many do, and it does lead to questions concerning the integrity of the Nightforce construction.
My own scope has been criticized in great detail in some of these threads, but it works for me, and I do not intend to change it any time soon. It is the Springfield Professional 6-20X56. There is not really enough vertical adjustment, however due to the mounting, the scope shoots to zero at 100 yards when the adjustment is bottomed out. The focus adjustment is on the front of the scope rather than on a turret, oh well, and I never use the illuminated reticle. On the plus side, it does has a mil dot reticle which has proven useful, but the best feature, and the feature which prevents me from trading the scope off is the "in view" level which appears just below the vertical wire. This single feature has improved my long range shooting more than any other. Yes there are levels which attach to the outside of the scope, but this one is right in your field of view, and you do not have to shift your focus away from your aiming point in order to see it.
If I was to purchase a scope today, and cost was not an issue I would choose between US Optics, Schmidt and Bender, and Nightforce. If I was budget minded, I would choose a Leupold, their fixed power target scopes are very nice, but a variable like the 4.5-14X would be better on days with thick mirage. Half the scopes I own are Leupold, and I am of the opinion when in doubt you can't go wrong with one of their products.