Here are my steps, you'll want to read up some on polishing metal, if you haven’t done it before.
Clean the metal, if needed. For this piece I used Mean Green & a stiff plastic brush. If rusty, soak in Evaporust.
This piece is plated metal. Based on this piece, I’ve concluded that you can go as far down as 400 grit and not harm the plating.
I use wet dry sandpaper (wet). I use it on my fingers and on a soft sanding block. For the sanding block I used pieces of Styrofoam packing cut to a sanding block a good size for the piece. I’d just bought something that came packed in styrofoam. A friend who polishes knives recommends cork sanding blocks. I haven’t tried them yet.
Wet dry sandpaper 400 grit, 600, 800, 1000, 1200, 1500, 2000, and 3,000. I finish with Mother's Mag & Aluminum polish.
Sand in all directions, and mostly in a circular motion.. You'll never get all the scratches out, but you can still get a mirror finish on many pinball metal surfaces. For the early grits, you can apply some pressure, but lighten up as you advance. Remember you're polishing, not sanding.
Wipe the piece with water between grits to remove the last one before stepping up to the next one.
Cut your pieces of sandpaper and soak the sandpaper in warm water. I lay out a flat Tupperware. Add just a drop of dishwashing liquid to break the surface tension of the water. I let the pieces soak as I polish.
After the sanding, wipe clean and gently buff in the Mother's Mag with a soft cotton or microfiber cloth. Use sparingly. At first, the cloth will get very black. Keep going until it eases up. End up gently buffing with a clean cloth. It isn’t going to look good until you get to this step.
You don’t always have to start at 400. A lot of times you can start at 1000 or 800. You'll learn with experience. I’m still learning. Start higher and see how it looks. You can always start again at a lower grit.
Best luck with your polishing!