Wax doesn't do anything, it just protects the paint. If your paint is currantly nice and shiney just wax. If it needs help well wax is like step 5 you are skipping a few.
There are various compounds out their, Griots has 4. Depending on how bad your paint is you may only need to use 1, or maybe 2-3.
Buff with a machine, hand polishing is a waste of time.
But even before you polish, you need to clay the paint. Step 1 is wash the car step 2 is clay the paint. After those two steps you are ready to go at the paint with a buffer.
Once the paint is up to spec apply two coats of wax.
Their is no fast way to bring tired old paint back, it requires lots of work. My SDL was pretty dull when I bought it and now the finish is a mile deep.