I would agree with paperslayer here. I think of a precision rifle as having two areas of competence:
1. Things that make the gun shoot better. i.e. must have fundamentals
-Barrel: chambering, rifling, contour and crown.
-Action: bolt face, headspacing, barrel threading
2. Things that help me shoot the rifle better. i.e good to haves human interface devices.
- Stock
- Scope
- Trigger
- Doodads (angle indicators, rangefinders, bipods)
Going all out on #2 items, while neglecting #1 will make you very good at shooting a poor rifle. The reverse is also true, where you could have a true precision rig that you can't shoot well because of the human interface. My generic advice is to get the best #1 you can afford, then decide on what to get of #2 that will maximize your use of #1.
Starting with a REMINGTON rig, your best bet would be a barrel and blueprinting of the action first, Lilja or Kreiger done by a good gunsmith. Then you can look at a jewell trigger and a very good scope (NF). Then the rest as you see fit. Your best bet is to go to .223 or .308 on a new custom action with custom barrel as you will have top notch fundamentals in a caliber that is inexpensive and proven and easier to learn to shoot very well. Then you can improve your human interface items as desired.