Wash with a shampoo not wash and wax..don't use a chamois to dry because you may as well be using wet and dry.
Use a drying cloth or silicon squeege...if you're going to clay the car don't worry about getting it too dry.
Claying the car has the same effect as using a paint cleaner only quicker..no need to paint clean after a clay.
Use a good quality polish..you could look at machine polishes, swirl removing polishes, etc.....but hand applied polishes to look at are AutoGlym, Poorboys and Meguiars... Polish not a polishing wax.
Wax the car to seal in the shine...poorboys blue wax, Meguiars Wax.
General rules...
Don't apply different stages of the clean with the same applicator.
Microfibre cloths are best for removing the residue...but don't scrub the residue off. If you have applied too much or it isn't dry enough it will gum up on the cloth and make it more difficult to remove.
Meguiars do a good 3 stage system .. cleaner, polish and wax. With the right applicators and cloths you can get a cracking shine.
Beyond that you're looking at machine application of polishes...ideal to get rid of all the spider webs in you paint surface.
A really shiny car is a labour of love..it could take you a full day to properly shine your motor..and the products will cost you a fortune...
but the satisfaction of knowing that you did it, has to be better than paying someone else to do it.