The advice is good on protecting the checkering. When you get the stock smoothed down with fine sandpaper, try wiping with a damp rag to raise the tiny whiskers of wood. Then sand them off.

I' ve used Watco Danish Oil finish on a couple of stocks, and it works great. It penetrates and then hardens to give a good, tough finish. I finished my stocks with spray polyurethane, satin finish. First sand between coats and then use steel wool as you get near the lasts coats. I' ve used as many as twenty spray coats to get the pores filled. When it looks glassy smooth under low angle light, you are nearly finished. A stock finished that way is really beautiful. Good luck.