To remedy the situation, I purchased the Meguiar's Headlight Restoration Kit at Pep Boys for $24.99 plus tax. I'd recommend also picking up some painter's tape to protect the car's painted bodywork around the headlight.

I considered buying one of the cheaper options such as the bottle of Turtle Wax headlight restorer or a generic brand buffing compound but neither one came with a buffing pad drill attachment.

The power drill polishing and buffing pad is absolutely required unless your car's headlights have only minor hazing. It would have taken me hours to achieve the results below without it.

The Meguiar's Headlight Restoration Kit directions recommended using a cordless drill for better mobility but I found that my old wired drill had a lot more power and stamina.

It also helps to hold the drill at an angle with minimal pressure for better control.

Pushing too hard or placing the buffing bad straight on the headlight just makes it very difficult to move it smoothly across the headlight.

I was impressed with how well the Meguiar's Headlight Restoration Kit was able to wipe years of haziness and cloudiness from the surface of my decade old headlights.

Unfortunately, some of the blemishes on the turn signal housing were impossible to remove.

It was also harder working on my car's headlights due to their narrow rectangular shape and several sharp plastic points.

I was able to buff the two Corolla headlights in about 10 minutes each using much less PlastX plastic polish since they only had about 5 years of wear on them.

Buffing away the cloudy appearance from the headlights made a dramatic difference in the apparent age of the vehicle.

If you're not feeling too proud of your older car, I'd highly recommend using a headlight restoration kit.

I was originally looking to buy a Mothers PowerBall Headlight Restoration Kit ($25), which includes two sandpaper pads to use on especially worn headlights.

I think that using the sandpaper pads would have helped cut down the polishing time on my Pontiac Grand Prix's 10 year old headlights.

Tips From Dennis Via Email -

There is a UV coating over the face of the headlight lens. Polishing the lens smoothes the plastic but also removes this protective coating (basically a clear coat).

The lens will be clear afterwards, but will eventually (couple of months) return to a cloudy appearance due to the suns UV rays clouding the poly.

Here's what I ran across and have used successfully for a few years now.

1. Wash the lens with soap and water.
2. Clean and polish the lens with Meguiar's G12310 PlastX Clear Plastic Cleaner & Polish.
3. Buff off the residue and wipe to lens clean.
4. Reestablish the lens protective UV coating with Meguiar's G170804 Keep Clear Headlight Coating.

No matter which method you use to polish the lens, the critical part is to reestablish the protective coating afterwards. This should be done annually (even on a new lens such as your aftermarket lenses) as the coating wears off due to what they are exposed to.
 

For more, check out all of my Pontiac Grand Prix DIY Repair & Maintenance Guides.