I've tried nearly all of the methods mentioned here, and many of the brands (but never Zaino's; I'll have to try that stuff, too - thanks!). I have a file folder full of sheets of sandpaper/emery paper of different grits, up to 3000, which feels just like very slightly rough paper. In fact, in certain situations I have used plain photocopy paper as a polisher, although you have to be very careful not to crease it; creases can scratch!.
MOST of the time, however, I fall back on Meguire's products for the bulk of the work. It fairly consistently gives the best overall results, often for the least work, of any brand I've used. Plast-X is simply outstanding at making clear-plastic things look NEW again, although it can take a long time.
First, remove any deeper scratches. Fine/gentle "Cutting" compounds like jeweller's polish, and/or fine-grade wet sanding are in order here. Don't use heavy grades (lower than 800 or 1000) for the first sanding, or you'll spend hours doing successive sandings until all the scratches from the sanding are sufficiently removed. When you are up to the finer grades (around 2000+), then switch to the polishes. If the scratches are only minor ones, you can skip this step altogether and save yourself a lot of work and hassle. Minor scratches come out with PlastX alone!
Sometimes I'll use 3M Automotive Plastic Polish first, though, to smooth things down a bit, because it is a little faster than PlastX, then finish with the PlastX.
Apply the stuff with cotton facial pads. I use two types: the small (maybe 1-1/2" square) kind that really feels like cotton, and the larger (about 3" x 4") kind that is a little more cloth-like. Put some the PlastX on the pad, then gently (it's as much chemical action as physical) rub in small circles over a limited area, perhaps 6" square. Keep rubbing and rubbing and rubbing over the same area, until the PlastX has almost disappeared. Then take a clean pad and buff it for a while longer, to remove the remaining faint residue. Repeat as many times as necessary, to get the depth of wet, glossy new-ish-ness that you want.
Note that especially on softer plastics, cotton pads themselves can often be sufficient to give a very good shine. Really worn, very clean fine cotton cloth (like from high-thread-count undershirts) can go even one step further, sometimes, if you make a soft little ball out of it (avoid wrinkles against the surface as much as possible) and use it gently.
For painted plinths, use whatever products to get deep filth off first if needed, then use Meguire's "Deep Crystal" paint cleaner for the final cleaning. Apply, buff off similarly with the cotton pads. Repeat a few times until you get that deep, rich, NEW looking shine.
Finish off with Meguire's "Gold Class" wax for a protective shine. Again, buff this with the cotton pads, then finally with soft cotton cloth (pieces of old --clean!-- cotton underwear work well, but NO seams!).
Similar approaches work well for metal platters: start with fine sandpaper to remove any deeper pitting, then use metal polishes like AutoSol. Finally buff to a shine, and seal with Meguire's "Gold Class" wax. Buff well.
For the rubber mats, use regular dishwashing detergent (CAREFUL! Do not use cleaners with a bunch of unkown chemicals, just plain detergent) diluted in water. Scrub very gently with a toothbrush if it is particularly filthy; otherwise just use a sponge or your fingers. Rinse very well in clean water, pat dry with a clean cotton towel and let air-dry a bit. Detergent solution does wonders not only to clean the rubber, but to actually restore a nice, soft and black finish to it.
Put it back on that shiny platter, above that shiny plinth, with that sparkling crystal-clear dust cover on top.
Enjoy the look for a moment.
Then make sure your tonearm and cartridge are set up properly (see relevant threads for details).
Then go do what it says in the vinyl-cleaning posts, and clean a record you like, very, very, very well, just like you did with the TT.
Sit back and enjoy.

You've earned it!
Oh, and one thing I forgot to mention: it helps if, as you get into the later stages, you wear clean, soft cotton gloves on your hands. Otherwise, every time you touch the thing to move it's position or just to hold it steady, you put fingerprints on it that you have to buff off again a few moments later! With the gloves on, such incidental handling is more likely to actually add to the polishing action!