You can use a toothbrush and stripper for the checkering. WEAR SAFETY GLASSES THAT ENCLOSE YOUR EYES. Sanding will flatten the tops of the checkering and make it look bad. A sand blaster will remove the soft portion of the rings of the wood and leave the harder parts raised.
OT, but hope to be of some help:
There is nothing wrong with varnish (I assume you mean polyurethane) except that it is difficult to apply well with a brush and can't be repaired when it gets scratched. Satin is easier to apply well with a brush, gloss highlights imperfections- and there will be imperfections.
Many people will tell you to oil it. Some oils are good, others aren't, and I rarely know what's in them. Also, there is quite a bit of myth out there about "hand rubbed" oil stocks after "sanding to 2000 grit." Generally, anything over 220 is gravy, I sometimes "feel" better going to 400 with oil, but it doesn't matter.
Another good choice for a finish that looks like oil is thinned shellac (thin w/denatured alcohol). It is easy to apply well and will appear thin and semi-gloss. You can use steel wool with mineral spirits to dull it to a satin, or an unthinned solution for a gloss coat (again, brushing gloss finishes is harder). If you were to polish the finish, it would polish much better than polyurethane (poly is rubbery or tough) and about as good as laquer- I doubt you will polish the finish (rottenstone, pumice and all that). Both laquer and shellac can be repaired, as they both contain solvents for their dried counterparts. Laquer looks good, but is too hard/brittle for a gunstock. For a semi-gloss shellac based rub on finish "padding laquer" is not too hard to use, but I would practice on scraps first- it takes a "touch." It can be applied over a sealer coat of thinned shellac. Just a reminder, alcohol eats shellac. If you're worried about knocking your Bud Light onto your gun and ruining the finish, the finish is the least of your problems.
If you brush on a finish, dust free and moderate temps are your friend. Too cold, and it'll dry way slow, leading to more dust in the finish and greater chance of runs. Too hot, and the finish can dry fast and leave brush marks that didn't have time to flow out.
I think thinned orange (unbleached) shellac on walnut looks very nice. Of course, you can do whatever you want- I just wanted to let you know a bit about what's out there and maybe save you some time and frustration. Wish I had an A Bolt, maybe in 243...