First, congrats on your new rifle!
A good way to save ammo when zeroing a new rifle is to bore sight it off of a bench with sold sandbags or one of the purpose made rifle rests.
You can't do it with all rifles, but with your new Remington 700 it will be easy to do.
Place the rifle in a steady rest and line up the rifle to the target with the scope.
Remove the bolt and then look through the bore as if it was a peep sight. Once you have the bullseye in the center of the bore and the rifle remains steady there, look through the scope. At this point it will probably be off by seveal inches or perhaps even more than a foot.
It will be easy to use your elevation and deflection turrets to bring the crosshairs to the center of the target. Remember though, your adjustments with the scope will be opposite of what you see. In other words, if the crosshairs are to the right, adjust your scope in the direction of "right" on the windage dial. We are adjusting the scope to the rifle and not the rifle to scope in this process, so opposite is the word here as it is opposite of your normal procedure. Right is left and left is right.
Once you have the rifle bore sighted, then return the bolt to the action and fire your first round. Fine tune your zero from this point and now your turret direction arrows will be correct. Right is right, left is left. Bore sighting isn't perfect, but it will get you close. I've used this with bolt rifles as well as our M4A1s and M16A4s when zeroing the ACOG optical sights. In the field, select the corner of a building, small but distinct rock on a hill, etc. The further away your target object is, the more precise the bore sight will be.
The old M67 90mm RR and the M40A1 106mm RR used to come with a sight plate (peep sight if you will) for the breach and index marks on the muzzle to place wire or string on to form a sight in order to bore sight the weapons to their optics. We are now using this concept to sight your rifle, but without the muzzle index aid.
If you have a semi-auto or other rifle that can't be bore sighted, then start your zero process at 25 yards. Any misalignment between the bore and the scope will generally be within the confines of your target paper and thus easy to see. Now adjust into the center at 25 meters/yards and then final zero at your prefered zero range. Remember though, that at 25 yards, it will take 4 MOA for every inch that you are off and not the 1 MOA that you get for 1 inch at 100 yards. In other words, if you are 5 inches off at 25 yards, you will need to move the scope 20 MOA on your scope's turret to move the strike of the next round into the center.
A third option would be to use one of the purpose made bore sighting units by Bushnell or other manufacturers.
Good luck with your new rifle!