I find "shining the copper" to have three parts:
first and easiest is the surface 'discoloration' / oxidation
second more difficult is light staining from boil overs / inadvertent spoon, hot oil/fats oxidation, etc contact.
third, and a real bear, is burned on black stuff.
mine is stainless lined, so it'll not go for retin & courtesy polish. I should probably instruct my will executor to send it out for "a plain old polishing" before passing it on to the next generation.....
pretty much any acid will take care of the first and some of the second.
BarKeeper's Friend is highly touted - oxalic acid is the active chemical ingredient with a very fine non-scratch 'abrasive' BonAmi has a similar non-scratch ultra fine abrasive.
it's effective against all three - but the burnt on stuff takes a lot of elbow grease. BarKeeper's Friend leaves a matte finish - no noticeable scratches or rub-grain marks, but it does not make a mirror finish.
old kitchen trick: ketchup. smear it on, let it sit 1-2 hours, rinse off.
or, ala how I "discovered" the trick works.... made a meat&tomato sauce for pasta, pot in sink, filled with water.
later a smaller pot got set 'inside' the tomato sauce pot.
after dinner 'discovered' the smaller pot was bright and shiny. well, the half that was in the acidified water....
lemon, pickle juice, vinegar - all acidic. all will make surface oxidation 'evaporate' leaving a bright copper color.
if you'd like to experiment, roll a lemon to loosen the juice, cut in half, rub the copper pot.... it's'mazing....
I use my copper&lids from stovetop to oven - the oven heat creates some interesting hues & shades.
BarKeeper's Friend and a mild effort wiping makes the 'copper colored' again with very minimal effort.
one thing on BKF - keep a 'fresh' supply - I've learned cans left/lost under the sink for a year no longer have the same 'chemical attack' power....