Deer?
I was in the same predicament, got lots of advice, and ended up with a fine rifle (Browning A-Bolt .270). But knowing what I know now, most decent bolt-action .270s will kill deer at relevant ranges.
All the main makes of rifles will have quite good out-of-the-box accuracy. Unless you're already a sniper, you'll probably not notice one that is 1/2 inch worse than another. They'll all kill deer very nicely.
The .270 has the advantage of a relatively flat trajectory, while still packing plenty of punch. The .30-06 has the same general configuration and powder load, but it pushes a bigger bullet, so the trajectory is a little loopier. Some of the bigger cartridges being proposed here (.308) are going to hurt your shoulder more, and you'll develop a flinch. They're not necessary for deer, but are excellent for Elk or Terrorists at long distances! ;-)
What I would recommend is to get a .270 bolt action that you like the feel of, and practice practice practice. I suspect you'll be most accurate with a gun that you are happy to use, and do use regularly.
P.S. - Optics can / should cost you almost as much as the rifle. Some people swear you should buy more scope than rifle. I have two scopes, and the better one makes a remarkable difference. It turns dusk into daylight and that's when the critters are on the move!
Whoops! I see in later posts that you included Elk. I agree with those recommending a step up, .30-06 or better. A .270 is still usable, but generally regarded as a little too light for elk.
Thanks for the words of wisdom. What kind of scopes did you get?
"The .270 has the advantage of a relatively flat trajectory, while still packing plenty of punch. The .30-06 has the same general configuration and powder load, but it pushes a bigger bullet, so the trajectory is a little loopier"
Not true. A 130gr .270 has a ballistic coeficient of around .380. A 180gr .30 goes over .450. This means that the .270 will bleed velocity at a faster rate and have more drop at longer ranges. Plus, the much higher secional density of the .30 caliber bullet will result in much greater penetration. You have to use a 150gr bullet in you .270 and drive it over 2800 ft/sec to come anywhere close to the '06. AND - the 30-06 is just getting warmed up with 180 grainers, you can buy factory ammo loaded with 220's that make the .270 look like the light game round it is.
308 is a shortened 30-06 developed to work in automatic weapons. It holds less powder and has less kick than the older 30-06.