Re: .308 vs 7mm
I own both calibers, both very effective for your requirements, in factory and custom rifles. As a beginner, I would recommend the 308. The difference in recoil between the two using a good recoil pad, and some weight in the rifle, isn't that great. It is inherently a very accurate round, and there is a large variety of factory ammo, including match grade, is available. You didn't mention if you plan on reloading initially, but if not, you will eventually. The 308 is a breeze to reload with a large supply of components readily available. For the goals you describe, there are factory rifles available that are very capable shooters out at 1000 yards. You would still have the ability to build off of them once you begin to develop your personal preferences which only come with experience. There are several new shooters at our club which conducts several long range events each year. It's surprising how well they compete with the factory Remington and Savage rifles that are designed for long range using Federal Match ammo in their 308's. I have seen several of these new shooters hold .5 MOA or better.
"Let us speak courteously, deal fairly, and keep ourselves armed and ready"-T. Roosevelt