What kind of wood is it? Is it natural or stained? What kind of gun is this?
I'm assuming it's quality wood and natural, not stained, here.
I'd strip it by sanding and you're on your own for cleaning out checkering. If there's any staining form oil, etc., search the internet for how to lift or minimize them. Most methods involve heat form an iron, etc.
To refinish, I've had great success with tru-oil. Really thin the first couple fo coats, per their directions to help it fill pores, then put on a couple of coats straight. It'll be very glossy, but you can lightly steel wool it to dull, if you want a matte finish.
Best thing is upkeep. If you scratch the finish, just wet a rag with Tru-oil and rub out the scratch. It'll completely disappear.
Best of luck.