Should I choose to do a job like this using sandpaper, what grade of sandpaper do I need and what grits should I start with and progress up to if I want to reach a martial arts polish? I don't really think I need to make the sword shine like a mirror, but I would prefer not to have to do this again for a while.......
Use 3M Sandpaper seems to give the best results and is easily available at Wally world, and better yet Autozone. Considering the sword I'd recommend 220 as the lowest just to apex the edge (will happen fairly quickly). Then move to 400 grit to begin putting a vertical the scratch pattern. Begin cross polishing at 800, then 1000. That will suffice.
Quite frankly a martial arts polish in today's terms means nothing more than sharpening the edge and perhaps roughly refinishing of the ha. Edge/niku geometry is often lost, or utterly destroyed.
Your goal as you are starting is to just get the edge to apex. Don't worry about the cosmetics.
PS: Forgot to mention. Following traditional methods are best. Sit on the floor with an elevated stone. I know what I am telling you.