There's no cut and dried true, perfect answer to your question. There are a lot of good explanations, and a lot of good intentions posted here. As well as a few jokes, ie. the 416
My own personal experience would be to say that there's nothing wrong with having a magnum as your first rifle. My first brand new rifle was a 7mm mag, I bought it when I was 18, and I still carry it every fall when I'm hunting. The reason why some would suggest not to get a magnum is recoil, I say BS to that. The rifle I hunted with the 2 years leading up until getting the 7mm mag was a Model 7 Remington in .308 and it had more felt recoil than the 7mm. Thats a fact. It was a different designed stock, with a short 18.5 inch barrel. The 300 does recoil harder yet, but its still manageable. Personally I'd go with a 7mm mag myself. Although there's nothing wrong with a .308, .270, 30/06, 7mm mag, and 300 mag. Ammo is widely available for all of them, in most rural areas you'll be able to find said chamberings in the local stores. My very first gun ever was a .303 Lee Enfield, that'll do the job for you too, and again ammunition can be bought nearly every where for it. To confuse the issue a little more, I've always said that if I could only ever have ONE rifle, I think I'd die jk, it would be a bolt rifle in 30/06. It may be boring, but it'll get the job done on anything I'd ever encounter.
In regards to some things you'll read about needing heavy bullets to kill bears, moose, elk, and other big animals as such...again, BS. With little doubt they'll work, but a 150 grain bullet from a .308 will kill just the same as a 165, or 180 grain bullet will, but its your job to put it where its supposed to go. If a bullet hits the vitals, or central nervous system it doesn't matter wether its a 140 gr, or 220 grain, its going to put the animal down. You don't need premium bullets, yes they will penetrate deeper, no doubt about that, but the common soft lead core bullet will penetrate well enough to kill any of the animals you suggested in a suitable hunting round. You'll be fine if you go with 140 gr and heavier bullets for the .270, 150 grain and heavier for the 7mm, and 30 caliber rifles will do what you want. They'll kill the big game you mentioned, but you have to do your part to. It doesn't matter what caliber you shoot if you can't hit where you're supposed to. Pick a rifle that feels best to you and shoot it regularly.
In regards to the Rem. 788, .308...that's a perfect rifle to last you a life time. Good caliber, good rifle. I like the .308, as a matter of fact I just bought one for a buddy's son as his first hunting rifle. When he turns 12 this March he'll be getting a brand new Tikka T3 in .308 topped with Talley rings and a Bushnell Elite 3200 3-9X40 firefly scope. I wish one of my dad's friends would have done for me when I was twelve
