If the paint is original, they you'll have to be careful with how much "compounding" you to to the old acrylic enamel paint. Which probably means an orbital buffer or an upper torso workout.
Once you get the basic oxidation/haze off, gently, then use some (what used to be Meguiar's Show Car Glaze") product to put some of the oils back into the paint. Put one application on and let it fully dry. Then buff it off with some good towels. A week or so later, do another application. Then follow that, a week or so later, with an application of a "straight wax" with no abrasives in it. Meguiars and others have some of these. Do NOT use one of the new hybrid ceramic silicon dioxide sprays/coatings at this time!
You might not get completely back to the original hard shine of the OEM acrylic enamel paint when it was new, but you can get close. It has to be done in steps with different, specific products, too. Some one-steps might claim to do this, but only for paint that is seeing oxidation for the first time in its life, not paint that is quite a bit older.
The glaze can bring back some of the color, which is a good thing, but it might be "all the way" back. By observation, fading will be more prevalent on the horizontal surfaces more than the vertical surfaces of the body. Just the way the sun hits it.
When I first started to take care of our '66 Newport in the mid-later 1960s, I got some of the Classic Wax paste, one of the premium cleaner waxes of that time. It got the white paint back to a hard shine quickly, but the car still didn't really "pop" until I went over the bright trim. THEN it looked "right". That clear-anodize-coated aluminum trim really shined, then.
Paint rejeuvenation can be a somewhat involved process. A lot depends upon the current condition and how much of the paint is left. Which is where you have to use discretion in removing only enough of the old, oxidized paint to make things shine again. STOP at the first signs of primer! In the case of my '80 Newport with the Frosted Teal Metallic factory color, it was too far gone to bring back, so a cleaner wax is all that would work on it.
Doing the sides of the car first can give it a better "street look", then adding a hard shine to the horizontal panels can fill in the situation, even if the color might still be a bit faded. Better to have a hard shine on faded paint than "primer bleed-through".
Everybody has their favorite brands of waxes and such, I just have had better results with the "old", original Meguiars (well before 3M bought them).
Just some thoughts and exoeriences,
CBODY67