There are stages to refinishing, some more drastic than others. My suggestion would be to go slowly in stages, checking as you go to ensure you don't mess something up.
Start with the wood. (stage 1).
Pull the wood from the metal carefully.
Use rags & alcohol to scrub the wood, keep going till you have no crud coming off the wood onto the rags. This will remove lots of stuff & probably lighten a lot as well.
Get some Boiled Linseed oil form a refinishing supplier, or Home Despot.
Pour out a
small amount, about a shot glass worth & dilute with the same volume of alcohol.
Pour & rub into the stock everywhere till it's all gone & let it sit till you smell no more alcohol (overnight).
Every day for a week rub a little BLO into the stock buffing hard to heat the wood. Wipe off excess & buff well the next day.
Finish buff with hessian, or burlap. Really go for it scrubbing the heck out of it for that nice semi-gloss look. That should do the wood & make it look really nice. Do another single coat once a month for the next year (without dis assembly) to really build up a nice look.
messing with the metal & finish(stage 2)
Look at the finish before you do anything. Is it blue, parkerizing, or paint?
If you have blue you'll need to do hot blue, cold just will not work well. Get all the stuff for cold bluing & go for it, if you know what you are doing, or send it out for a professional job.
If it's park the same applies, just make sure you have someone who actually knows how to do these things.
Paint?
Oh boy..............
The paint is not available any more & was a specialty paint called "Suncorite". Probably the toughest paint ever applied to anything, & usually applied over parkerizing as well.
A friend has had good luck with "guncote semi-gloss black" as a duplicate for the finish, but not for the durability. It does need the bake on to work well though.
Whatever you do
be aware you will have devalued the rifle even though it will look better. Collectors want the original finish, preferably with "patina" (a "perlite" term for old, dirty rust) Hey it's your gun so it's your decision. If you want a collector don't do it, if you want a shooter well..............
Here's my savage after doing the wood as detailed, but leaving the metal alone, apart from oiling.
