This article is by request. There is a lot of information out there on how to setup and sight in a scope. I'm going to cover the basics.
The first step is to buy the proper scope for your rifle.
The general rule is that you should spend as much on the scope as you did on the rifle. I have $200 rifles with $400 scopes, and I have $3000 rifles with $1200 scopes, so the rule is very flexible. The point is to NOT put a $200 scope on a $3000 rifle.
The next step is mounting hardware. Get good bases, and rings. Weaver, Vortex, and Warne work well.
Putting the rifle in a cradle makes life much easier. I also recommend getting a torque wrench unless you are a mechanic and have one built into your arm already.
Get the bases on and tightened, place the rings on loose, take the ring caps off, and place the scope in the rings. Figure out the eye relief by getting behind the rifle. Close your eyes, get a comfortable hold on the rifle, and open your eyes. Move the scope to a position that feels right when your eyes open.
Place the rings as far apart as you can while still giving you some room for fine adjustment of the eye relief. Put the caps on loose, and slowly get them snug allowing the scope to move with a bit of force. Make sure the gap between the caps and rings are even on both sides. When tightening the screws/bolts on the caps, tighten them in a "X" pattern slowly.
Now that the scope is all snug but you can still move it, make sure it's level. There are tools you can use, but I prefer to "eye" it. It's great to mount the scope perfectly in line with the rifle, but not everyone holds the rifle level. If you are mounting the scope on a precision rifle spend then time to get this right. If it's a hunting rifle, mount it level as you hold the rifle. Just like the eye relief, close your eyes, hold the rifle like you are going to shoot it, open your eyes, and make adjustments. The trick here is that if you don't know which way to rotate the scope, then it's probably setup right.
The next step is to bore sight it. A laser bore sighter comes in handy. Place the laser in the bore and adjust the scope so the crosshair in centered on the dot. Use the longest distance you can. In the store I have about 20 meters and it gets it pretty close at 100 yards. If I'm at the range I'll use the laser at the distance I'm shooting.
With a bolt action rifle (or any firearm where you can remove the bolt) you can bench the rifle and look down the bore. Center the target while looking through the barrel and keep the rifle still while you move your eye up to the scope. Adjust the crosshair as close as you can to the center of the target.
Now comes the fun part. By "the fun part" I mean the easiest or the most frustrating thing you've ever experienced.
I'll be an optimist and assume easy.
Get your rifle in a good rest, something that will keep the rifle still without holding it. Chamber a round, line up the crosshair on the center of the target, and squeeze off a round. Luckily, there will be a visible hit on paper. Now place the crosshair on the bullet hole and keep the rifle still. Turn the scope adjustments until the crosshair is on the center of the target. If you feel the rifle move, stop, and start over.
If you don't have a rest that keeps the rifle still enough. Then we'll have to "walk in it". Figure out the adjustment steps of the scope. Most are 1/4 MOA, some are 1/2 MOA. With 1/4 clicks, 1 click will move the point of impact (POI) a quarter inch at 100 yards. That means it takes 2 clicks move the POI a quarter inch at 50 yards, or half an inch at 200 yards. That math works great when you are turning the dials the right way, so always make small adjustments.
The top dial on the scope with have a "U" or a "D" with an arrow indicating rotation. This means turning the dial in the "U" or Up direction moves the point of impact Up. The side dial is the same, it will have a "R" or "L" and an arrow.
An example would be a scope that has an "U" rotation of clockwise, and a "R" rotation of clockwise. There are 1/4 MOA adjustments. We fire a round at 100 yards and it's 3" low and 4" to the right of center. 3 inches, in 1/4 clicks is 12 clicks. We will turn toward "U" clockwise 12 clicks. 4 inches, in 1/4 clicks is 16 clicks. We will turn away from "R" counter-clockwise 16 clicks. The next fired round should be close to center. In reality, I would only move half the number of clicks to make sure the POI is moving in the right direction. The other option is to shoot 3 or 5 round groups and move the group to the center. If the rifle or ammunition has an issue you might end up chasing the rounds in circles. If you start getting frustrated or it's taking more than 20-30 rounds to sight in the gun, continue reading.
If the first round doesn't impact paper, if you had a round on paper and now it's it not, or if you are chasing the impact around in circles then you are losing it. This happens to us all.
There can be many problems. The scope may not be holding zero, the rings could be loose, the rifle or ammunition may be inaccurate. The first thing to do in bring the target closer. If you can't hit paper at 25 yards, then there is something seriously wrong. Start back at the beginning of this article. Get the rifle bore sighted again and get an impact at 25 yards.
Shoot a 5 round group at 25 yards. This should be slow, relaxed fire, with proper breathing. At this range the group should not be larger than 1". Adjust the scope to place that group in the center of the target. Now move the target to 50 yards and repeat the 5 round group. The group size should still be around an inch for most rifles. Keep moving the target out to the desired range.
If your 5 round group is bigger than one inch at 25 yards, there is a problem. Try a different scope, try some other ammunition, and make sure everything is bolted down.
If you have any questions or input let me know. firstname.lastname@example.org