I would suggest the following cartridges: 338 Ultramag, 300 Ultramag, 338 Lapua, 338 Edge.
You may also be OK with the following: 300 Weatherby, 300 WSM, 300 Win, 338 Win, 7mm RUM, 30-378, 338-378, or 7STW
I would suggest one of the following actions: Remington 700 (the 700P or Sendero should both be available in 300 RUM), Savage 110 or 12, Sako TRGs-M995 (338 LM or 30-378), Tikka M695 if you can find one or a T3.
The 338 RUM in a Remington 700 seems to be the most practical route, as you would be 'ready-to-go' from the start. You could go with the Remington 700 and start building from there.
I went the Sako TRGs/338 Lapua route because I found a decent deal on a TRGs in that caliber. Once I got over the cost of purchasing the brass, there really wasn't much of a difference in the actual shooting cost over any of the other comparible calibers, and I'm still on my first 50-pieces of Lapua brass 2 years and hundreds of rounds later.
The trigger in the Sako was nearly perfect so no changes were necessary there. I hunted with the rifle 'as-is' the first year. Then added a new stock (McMillan A-5 adjustable ~$700) between hunting seasons for the next year. Then had GA Precison pillar bed, skim bed, and add an aftermarket recoil lug to the stock for the next year (again between hunting seasons, ~$400), as well as lapping the factory barrel with Tubbs FF bullets (that was definately money well spent, ~$30). Before next season I plan to rebarrel and add a muzzle brake.
During the years of building, I have taken 2 nice mule deer bucks (one at 120 yards and the other at 505 yards), One elk at 400 yards, and a nice antelope buck at ~480 yards.
The estimated cost per shot is ~ $0.40 - $0.50 per bullet, $0.20 - $0.25 in powder, $0.03 primer, and $0.15 for brass so far (~10 reloadings). I also have some 200 gr Hornady bullets that I picked up for less that $0.20 each that I practice/play around with.