First, it depends on the metal and its condition.
In Pinball, many pieces are Nickel Plated, Stainless Steel, etc.
Once these are cleaned, one can buff the metal, but it wont replace plating, so what you buff will dull if the plating is gone, unless clear coated.
Youll want at least a medium size motor, face mask, stitched and light stitched or half stitched Muslin wheels.
Compounds are Brown - Course, to Green for High Polish....There is white or other colors mentioned for stainless, if you wish to try. Avoid Red at all costs.
An Ultrasonic helps immensely.
Harbor Freight is great for non professional, and they have a pack of wheels and rouge blister packed,
along with affordable motors and Ultrasonic.
Around $100-$150 for everything you need.

Consider Everbrite Coating on Buffed pieces not in contact with Ball, to keep shine, if one has buffed off the nickel plating.

To get a mirror shine, you need to look at the deepest scratches, and choose the right grit to remove enough metal to the scratch depth.
Then you need to move up the grit ladder, as far as you are OK to go, before you can polish.

You can skip the super fine sanding and go right to cutting tripoli, but if you cant hold perpendicular by hand, youll need a jig, to avoid fine waves.

For flat objects, sometimes easier for some to anchor flat, and wrap emory in block of wood to ensure
smoothness.

Same as wood, just different paper.....The finer the grit, the less work on the wheel.

Try the order listed...green to finish, although personally, I have not tried Blue.

Please avoid red (especially Chinese/overseas)....it can be toxic, and a pain to clean off.

If a piece of metal does not have heavy scratches, but needs cutting, start with brown on stitched.
The courser, gritted wheels are abrasive and used for deep scratches, pitting, etc, before getting to cutting and polishing compounds.
One should use a different wheel, for each color.