Match Rings and Bases
Virtually all modern rifles are either pre-drilled and tapped for scope bases or grooved for the appropriate mounting attachments, so the first step is simply to make sure that the mounting system you select is the proper fit for your rifle and that the rings are of the right diameter and height for properly positioning your scope. Certain types of scope rings fit only certain types of scope bases, so make sure the components match. Scope rings also come in different heights, which might cause confusion.
It's usually best to mount the scope as low on the rifle as possible without the objective bell touching the barrel, and allowing clearance at the eyepiece for free operation of the bolt. Ask a savvy gun shop operator for advice. Most bases and rings are attached with screws by using socket heads or Torx wrenches.
Begin attaching the bases by wiping the attaching surfaces dry and clean and then applying a thin coat of oil or rust preventive. Check that the bases are properly aligned and that you're not attaching them backwards.
Lock 'Em Down
For maximum security, apply a small drop of a thread-freezing compound such as Loctite (the semi-permanent Blue is best for scope mounting). Ensure an even fit by alternately tightening the screws. If you use rings that attach to the front base by a rotating socket system, do not use the scope as a lever to pivot the ring into position. Use a wooden dowel or something of similar diameter instead.
With the bottom half of the rings in place, position the scope and tighten the top halves of the rings only enough to allow the scope to rotate and move back and forth. With the rifle held level, rotate the scope until the reticle is precisely vertical and horizontal.
Adjust Eye Relief
Position the scope forward enough to ensure your eye is safe from recoil. As a rule of thumb, move the scope about an inch farther forward than you think it ought to be.
Double-check the scope's position and tighten the ring screws. Alternately tighten the screws for even spacing between the ring halves.
Insert a bore sight using the proper insertion pin for the caliber you're using, then adjust the vertical and horizontal axis to your desired point of aim.
Note: Bore sighting is only a means of getting your rifle to print on paper at 100 yards. To properly sight-in your rifle, you'll need to fire test groups at the range.