As a relative beginner it is not so long since I was faced with the same problem, though as problems go it is a good one to have. For a first rifle I found that 308W was a good choice for lots of reasons.
Firstly, as others say you need good glass however the "spend as much on the glass as the rifle" formula doesn't work. In my view one of the best stalking scopes for the UK is the 8X56 S&B and I got mine 2nd hand for £250. So, forget those who encourage you to spend a fortune on glass and instead try and get some top notch glass that is good value for your money. MacLeods of Tain get a great reputation and usually have an ever changing supply of 2nd hand glass and I think it is best to phone them. If you are not doing much low light work a 6X42 would work well and you don't need a variable magnification though you might want to have it. Prices have gone up a little since then but even so with careful shopping I'd say you could have absolute top notch, confidence inspiring, glass in the £300 or a bit more region.
In terms of rifles when people heard I was looking one for stalking almost everyone I talked to pointed me at a good weather resistant rifle with a plastic stock and the Blaser name came up quite frequently. I wanted a nice wood/blued rifle but in the end I went the plastic route as advised and with hindsight I'm glad I did. The price of Blaser has gone sky high and you may not want to spend that sort of cash, though 2nd hand ones do come up and in 308 it is unlikely you will ever see a shot out barrel in a Blaser as they are known for having very hard steel and for being very rust proof. The Blaser isn't fancy, but it breaks down to clean after rain, doesn't rust as easily as most of the competition, is easy to travel with because it breaks down into a brief case almost, easily converts to left hand use if that is important, has a great trigger out of the box and is known to be accurate and reliable. If this is your first rifle ever then the boring reliability and consistency of the Blaser, and the Blaser scope mounts, mean that if you are not getting decent groups or hitting the target then it is almost certainly something you are doing wrong. Put a good 2nd hand Swaro or S&B n top and you are totally certain it is something you are doing wrong
Should you not fancy a Blaser (and many people quite reasonably hate them) there seem to be a lot of good rifles out there with the Europeans tending, in my very limited opinion, to have the edge over the US made rifles in terms of quality and "out of the box" usability. I also believe that Howa, who I think are Japanese, get a very good name.
I hope that has maybe given you a few ideas and I will post a pic of my rifle as that might immediately let you know that Blaser isn't for you