Ok so every man and his dog will give you different techniques for polishing your car from putting the polish on the car body work and rubbing in a circular motion to pouring it on a cloth and rubbing in straight lines.
Firstly this is a how to polish your car by hand guide.
The range of polishes out there is immense from Auto Glym Super Resin polish and Menzerna Intensive Polish to awful cheap own supermarket brands for 99p.
I have always been a firm believer that you get what you pay for with most products, so I would personally stay well clear from cheap own brand polishes.
If you have new car its still worth polishing “well its new it doesn’t need polishing” wrong. It’s been given a quick wash at the dealership. Chances are though it’s been sat in a huge car park for a good few weeks.
Ok so you’ve washed it and I would now personally clay the car although the car looks clean there might be tiny bits of contamination still on the car from transportation or where its been parked.
Now all that’s done have a good look around the car for any imperfections if you haven’t got any congratulations your one of the lucky ones.

If this is the case you’re only going to need a very fine polish / glaze with a very light cut, something like Sonus SFX 3 Final Finish Polish or Chemical Guys EZ Glaze with Acrylic Shine II. Either of these will enhance the gloss and depth of your paint but will have little impact on defects.
I would use a Handi Grip Applicator and work on small areas at a time working the polish till its starts to dry to a haze or starts producing dust. Moving in a circular motion and buffing off with a Microfibre towel.
If your car does have some Random Deep Scratches (RDS)  you can remove these by hand by using a polish with slightly more cut to it, but be careful if in doubt have a look for a professional valeter or a car detailer as they will have a lot of experience in this area and will be able to offer advice or to remove them for you.

One of the safer ways would be to fill the RDS with a polish with filling ability’s, one of the best I have found is Auto Glym Super Resin polish commonly referred to as AG SRP. If you fill a RDS it hasn’t gone for ever it’s just a temporary solution that makes the RDS less noticeable to the eye, and make sure that you put a wax over the top of your freshly polished paint work to help seal in the gloss and depth and keep the look for longer.
When filling a RDS work over not along as it will fill quicker even small 2” RDS could take five to ten minutes to fill, with a few passes of Auto Glym Super Resin polish.
Each time you have worked the Super Resin polish buff with a microfiber and then check your work and just keep going until it’s less noticeable to the eye.

Ok polishing your car by hand is a long process even longer if you’re trying to take a lot of defects out. If you have a lot of RDS and swirls you may want to look at a machine polisher such as the Meguiars G220 you will remove defects a lot quicker and the overall finish will be of a higher standard than polishing your car by hand.

Mercedes Headlight Before Correction

During Polising

Mercedes Headlight During Correction

After

Mercedes Headlight After Correction