Good starter rifles with decent price tags line the new and used racks of nearly every gun store in the United States. A .308 bolt rifle with a decent scope is an excellent choice. Remington 700, Ruger M77, Winchester M 70, Weatherby Vanguard, Browning A Bolt, CZ in various models, all top shelf. Sako on top of the top shelf. Savage is darn good too for the money, same same Stevens and Tikka has a following too. Milsurps are fun too and cost effective for the budget minded.
By looking at lots and lots of rifles you can get an idea of how different stock makers dimensions feel to you (fit is very important for comfort in recoil and accurate field shooting), how the bolt feels, where the safety is located and how it works, how the magazine is set up (blind box, hinged floorplate, removable box) and how you like that aspect of the rifle, see different styles of recoil/butt pads, different scope mounting systems, different open sights if that is important to you and generally learn a hell of a lot about what you think you really prefer. I think that hanging around gun shops and looking at lots of rifles is the only way to buy another rifle, especially if you're just getting started. For scopes, Leupold, Bushnell Elite, Nikon Monarch, Burris Signature, Sightron SII and on up are in my decent scope category. IMHO a good rule of thumb is to spend about the same on glass as you do on a rifle until you get in the $1000 plus on a rifle territory. A good mounting system (rings and bases) should not be overlooked either.
For what little it is worth, I think getting a rifle and scope you really like in any adequate chambering is a lot more important than fretting over what round the rifle will shoot. There is such a huge selection of chamberings available today with often minimal differences in actual performance that what chambering the rifle in is often secondary. I think in terms of light centerfire, medium centerfire, heavy centerfire and African. Light being the .22 rounds up through maybe .243, medium starting at .243 and going through the many .30 calibers and heavy going up to the stuff that is .375 H&H or approximates .375 performance. African to me is .375 and above.
For paper punching and zombie duty, anything in the light or medium centerfires will get it done. Popular choices for the best round will abound and you will be able to read a lot of folks suggestions as to what you should buy. My suggestion is to buy a rifle that you like and that fits you well.