I polish lids daily. I restore turntables constantly. I really enjoy it. I have the polishing process down to 1 hour.
1) assess the damage and start with the appropriate grit of wet sanding paper. I use soapy water and a black and decker mouse. Just a light skim of soapy water from a spray bottle.
2) 800 through 3000 - overlapping circles and then crosshatch grid.
3) mcguires ultimate compound on wool buffer
4) mcguires scratch x 2.0 on finishing foam pad on buffer
5) carnuba wax on ultra soft random orbit pad
6) hand buff off residue with flannel cloth or high quality micro fibre....
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To do the inside or just the sides....place your buffer upside down in a vice and hold the lid in your hands and hit the front and sides....interior can be tricky and use caution as to not hit the inner sides with the edge of the polisher. Sometimes you will need to repeat the process, but more often than not it will be a huge improvement.
Cigarette burns, melt lines from cords stored on top of the lids are my specialty. One time I had to start at 100 grit, because the original owner went for a frosted look with 220 standard paper....lol.
I usually blaze a fat spliff and enjoy the process.