First step up from what you have is a good brush gun, and considering your location, it is hard to beat a .30-30 lever. It is undoubtedly the most popular in history. Great varmint gun, good for deer up into the 100 yard range. I prefer a .45/70 lever, but here in the Rockies, a little more poop is nice because of big bear and moose, etc.
Next would be a long range rifle - Like many here, I would recommend a Remington 700: 30-06 is fine, as many have recommended... I shot a 7mm Mag for years, and now have a 300 win Mag, which I will stay in till I die...
But a good brush/saddle gun is more important. Scoped rifles are good for their purpose, but that purpose is 125+ yds. They generally suck in the woods, where it is uncommon to get a range of more than 100 yards, and often down to 50yds, where a scoped rifle is just about useless, and high power will cause deflection...
As for caliber, I prefer .30-06 because of its versatility. You can go up to Maine and hunt moose, hunt black bear locally or go out west for elk or antelope, and the ,30-06 will do everything you need it to do. It can stabilize heavier bullets and will give a flatter trajectory than the .308, especially if you handload. But if you do not envision ever hunting anything bigger or more dangerous than a whitetail, then a .308 will fill your needs just fine.
If you are putting more of an emphasis on self defense than hunting, then a Remington slide-action rifle like the Remington 760 may be a better choice. You sacrifice some accuracy and reliability, but you can get ten-round aftermarket magazines that give you some pretty good firepower. A 760 in .308 would not be a bad choice.
So my advice is use the 870 as your primary self-defense tool, and get a Savage 110 in .30-06 as your hunting rifle and long-range defense rifle. I recommend the Bushnell Elite 3200 3-9x scope. It is an excellent buy, and the Elite series are far superior to previous Bushnell offerings; damn close to Leupold in quality at a far, far lower price. I think this will give you the best value for your money, and the most flexibility.