How much it costs depends on how much you want to do - it may well be less expensive to buy a used match rifle... Worth considering.
A barrel - buy a good one from Krieger, Hart, Lilja or one of the many other high quality barrel makers. Your smith may have a barrel maker he prefers to work with. I'm a Krieger fan, but there are many other very good barrels available. Figure $300 for the barrel, then you're going to have to pay the smith to true the action, chamber and thread the barrel and also cut and crown it. I'd estimate around $500 - $600 all up for the new barrel by the time it's installed. Krieger, Pac-Nor and Hart will install the barrel for you if you send your action to them. The wait is long, but workmanship tends to be good.
Stocks - for starters find the McMillan web site and take a look at different competition style stocks. Figure if you're going to shoot benchrest, or NRA Highpower or F-class, which is a subset of NRA prone competition. You can shoot a scoped rifle in either Highpower (any rifle/any sight) or in F-class. Otherwise highpower shooting is done with very nice sights, more expensive than many scopes. Benchrest is of course done with a scope. The type of competition/target shooting you want to do will have a direct influence on your choice of stock. Stocks made for benchrest are not particularly well suited for highpower and vice-versa. Decent target stocks go $500+, unless you catch a good used one for sale. If you don't want to do the bedding work yourself, figure another couple of hundred bucks for a smith to do a good job of pillar and glass bedding it.
Cartridge choice. Since your rifle is a .30-06 already, the bolt face can accommodate a wide variety of options. If you're not hunting with the rifle, POWER isn't really a factor. You don't need to penetrate an elk shoulder at 800 yards, just get through paper. What you want is ACCURACY! Starting with the .30-06 bolt face, I'm hard pressed to come up with a more accurate cartridge than the 6mmBR or the slightly modified cases based on the 6mmBR, such as the Dasher.
6mmBR Cartridge Guide
6BR Improved Cartridge Guide
No - you don't need a bigger cartridge to shoot well at long range. A step up to 6.5 or 7mm could work well too, particularly with something like the 6.5-284, which could be built very nicely on your pre-64 Winchester action...
You've got all kinds of options with a pre-64 Win Model 70 .30-06 to work from. It's not likely to be inexpensive turning it into a long-range match type rifle though. Frankly, I'd seriously consider selling it, and shopping for an already built match rifle. If you want a new, ready-to go rifle, look at the Savage match rifle avail in 6.5-284 right from the factory... They're really pretty darned good.
Best of luck! Guy