I have seen this entry on the forums all too many times but I do love throwing my two cents in! I went through this same exact dilemma.
First, you need to consider the optimal caliber. Personally, as versatile as the 308 is, I believe there are many other choices that are far better for what you trying to accomplish. Second, and more importantly, if you are planning on shooting out to 1000+ you need to spend more on your scope. Glass is by far more important than your rifle. Rifles can be tweaked, upgraded, finessed but a scope is a scope. Save your money and buy quality glass. Trust me it will make a WORLD of difference.
OK... onto the rifle choice...
There are tons of notable makers to consider, and everyone on every forum will tell you which one is the best and why. Which is exactly what I'm going to do haha. Personally... IMHO, I believe Remington and Savage are two of the easiest, most common, most reliable, and most upgradable firearms to own. They are also proven to be quite accurate "out of the box." The first Savage I owned shot clover leafed holes at 100 yds with it's first shots. Can't complain about that.
OK, so you said you want a Savage 11/111 FCNS. First... 11 is short action and 111 is long action. Honestly both do just fine. Short action will allow faster cycling of the round, long action will have added strength. With that in mind either work find. I own a 300 WM that I opted for over the 300WSM because I felt the cartridge had ballistic advantages over the short action that far outweighed the loss in speed of cycling the next round. In your case it's a horse a piece. However, as a target shooter, a 308 is virtually archaic. Don't get me wrong, there are plenty of gunny's that can probably shoot the nuts off a fly at a 1000 yards with a 308, but there are so many other calibers that make it much easier to do.
308Win is possibly the most common cartridge available. It is quite versatile and has it's place in target shooting as well as hunting. However, there are literally dozens of calibers that I would pick over the 308win. The 308win is what I consider a "middle of the road caliber." It is "good enough" for long distance targets, and "good enough" for short range hunting (long pending on the game you're shooting). With that in mind, if your primary concern is target shooting, you will most likely be spending quite a bit of time at the range firing quite a large number of rounds. Consider this, as a hunter, you want the most amount of stopping power needed for an ethical kill on your game. That equates to a sore shoulder at the range. Let me reference my 300WinMag again... this is an excellent cartridge for long distance shooting. It is the standard sniper caliber of the US Army, and has phenomenal stopping power past 1000 yards. However, on those days that I just want to fire a few dozen rounds at some paper, I come home with a bruised shoulder. My rifle kicks harder than a 12 gauge. The issue that brings up to a new shooter is the strong possibility of developing flinch. This is really bad.
What other than the 308? Well, again, that all depends on your shooting skill/experience, intended game, and intended distance for both targets and hunting. If all of your long distance shooting is strictly against paper, than you do not need something like a 338LapMag. You want something that will be comfortable to shoot and not break the bank when buying rounds. There are lots of people that shoot 22's to that distance and that is about as cheap as it comes to shoot. However, if you are hunting past a couple hundred yards you better be a **** good shot to take down that game. So if you plan on doing a bit of hunting I would opt for something larger.
So what would I recommend? Drum roll please...
6.5x284 Norma is IMHO the best caliber for just about everything. There are a fair amount of reputable and quality ammunition manufacturers for a fair price. The 6.5x284 dominates in 1000 Palma competitions. It has very mild recoil and is supremely accurate. Plus, because of the longer than average bullet length there is a high concentration of mass into a smaller point of impact... better known as sectional density. With this in mind, it will shoot faster and retain its speed better, be less affected by wind/bullet drop, and hit with more force than the 308 allowing it to penetrate much deeper into the animal... thus equating to a better kill.
Savage makes several 6.5's. The Creedmore is amazing as well. I just personally feel the 6.5x284 Norma has some advantages. Many of these middle wildcat cartridges are considered the only options for long distance target shooting. Even the 6mm is winning competitions.
As a new shooter, these calibers are a breeze to shoot and are phenomenally accurate. The downside is that many of these rifles are 700-900 just for the rifle. Savage has several.
A backup to that choice is the .260Rem. Similar ballistic coefficients (rating that determines ... in short explanation... how well it fights the wind. higher the number the better).
Honestly the 308 is a great choice for something that will serve you well and be readily available. You also have to consider how often you will realistically be shooting out to 1000 yds. As a new shooter, 1000 yards is a daunting task. Consider this... if your rifle is a 1-MOA shooter, that means it shoots a 1-inch grouping at 100 yds... the same rifle will shoot a 10 inch group at a 1000 yards with NO WIND. Now add the fact that wind is inevitable... and consider this... if a crosswind pushes your 308 bullet 2 inches at 100 yards horizontally, that means it will push 4 inches at 200 yds, 8 inches at 300, 16 inches at 400, and 32 inches at 500 yards (roughly). With that in consideration... you have to account for 3 FEET of wind already... at HALF of your distance.
My advice to you, is find a caliber in your budget that is realistic to shoot. Practice practice practice at your local ranges. Your best bet is to work at 200-300 yards until you are supremely confident in all directions and speeds of wind. Use different loads of rounds. Heavier/lighter 308's will fire differently as will the type of tip. Is it rounded for hunting, is it pointed for target shooting? Study the ballistic coefficients, purchase a ballistic software and learn how your particular caliber, with the particular weight bullet, will drop at relative distances, how wind will affect it. Get proficient at target distances up to 500yards. When you are a viable threat on the range, feel free to upgrade to the long distance shots like 1000 plus. I bet by then you will be looking at another rifle and can afford your target rifle/scope.
Soooo.... I know I've rambled a TON. And what it all means... is that you need to find what will suit your budget first and study the calibers. Study your scopes. It is by far more important to budget your money for a quality scope. But, if you are just getting started, a 308 will certainly do the job until you are ready for the 1000 yard targets, or long distance hunting.
Hope this helps. Feel free to PM me with more questions.
TK