Sanding dust mixed with dilute finish on the stock is a great way to go. About #320 wet/dry paper will work well. Dilute your finish of choice with mineral spirits and wet sand the stock. Let it dry, repeat if needed. Plain old spar varnish works great for this.
Filling the grain can be done with the finish. It takes many, many coats. It is a pointless exercise. When you hear about some one needing a gillion coats to finish a stock that is what is going on. There is no magic or desirability to it. It is just doing a simple job the hard way.
I also sometimes use grain filler. Get the walnut tone or black, Woodcraft has it. Finally, Deft sanding sealer will do the job quickly. It dries real fast and sands super easy without cloggind the sand paper. Several coats of it, sanding between coats, will do the job.
Once filled use whatever finish you want. Half a dozen coats or boiled linseed will look nice and old time-ie. Smells good too.