Since you're going from .243 to .260, you'll need a new barrel. Great -- that will make more difference than anything else you could do to improve accuracy. I've had good luck with both Krieger and Hart, but winning equipment lists often include Broughton, Lilja, Bartlein, Shilen, etc. When you find a good smith you can trust him to choose a barrel maker if you like. Finding the right smith (assuming you're not doing the chambering yourself) is the most important thing to do first. Figure $500-600 for the barrel and chambering/fitting. Pick a barrel twist and freebore optimized for the heaviest bullet you plan to use. Your smith can help with this, too.
You may want to have your action trued when it is barreled. A good truing job will cost around $250. Frankly, if you'd be happy with 0.5MOA you can probably skip the action truing.
I don't mind carrying a 10-pound rifle for hunting, so I usually use H-S Precision stocks and a Rem varmint contour barrel. They're easy to bed yourself, because the aluminum bedding block is already a fairly good fit. If you wait for them to go on sale, or buy someone's take-off, you can often pick one up for under $300.
Of course, you could also pillar bed the stock you have now, if you like the barrel contour in your .243. That would save some cash.
The Rem trigger, properly adjusted, is perfectly fine for a 0.5MOA hunting rifle. If you want to go top shelf and spend a little over $200, the Jewell trigger can't be beat.
In my experience, the three things that matter most for accuracy are bullets, barrel, and bedding (in that order). If you shoot bullets of proven accuracy potential (Berger would be a good place to start) in a good barrel with proper bedding it will be hard to shoot a group bigger than 0.5MOA in good conditions, if you do your part. You'll want a top quality scope, good quality reloading dies, and good brass (you can make .260 from Lapua .243 or .308 brass, which is what I would do).
Check out the 6mmbr.com site -- good info on projects like the one you're considering.