Both are very similar to each other. 30-06 Springfield will do the same thing as 308 Winchester, but 100-150 FPS faster. For rifles go to your local gun store and handle a few, put them up to your shoulder. For where you are starting at I'd suggest looking at packaged rifles. Remington 783 in the wood stock is good (Comes with the Vortex Crossfire II), Savage Axis II (better trigger, better scope, scope will vary between a Weaver and Bushnell, both aren't life time scopes, but they will get you through a season or two.) Savage 110 Package rifles are a good option too. I've really been happy with my Weatherby 223 and the Vanguard packaged with the VX-2 is slightly above your price range but deserves a look. The biggest drawback for a lot of people is the hinged floorplate, but if you are walking a lot it won't be a huge issue.
Rifles are such a personal preference go out there and look and handle. It's hard to buy a truly bad gun; the internet is such a doubled edge sword that guns that have huge mechanical issues don't make it to long. Quality control and other related things are another matter. When you buy a new gun make sure to look it over before paying for it. And keep in mind on the internet and forums everyone has had a bad experience with something and if you ask that question a lot of people will use that opportunity as a sounding board for their own frustrations. And as I always say, if you buy it and hate, you can always sell it and get something else.