The other thing to consider, of course, is that at least some of the marring that is present would certainly have been present on the car when you picked it up. New cars are never perfect, so don't take all the blame.
Use of a quality wax and/or sealant will help to protect the finish. Something like wolf's body wrap, which is a synthetic paint sealant, would be a good place to start.
If you want the marks corrected out, have a chat to Blake from Envious Car Care - he's up in your neck of the woods and does sensational work.
auto detailing the Hunter - he could even set you up with a correction and permanent ceramic coating package which will set your car up for the foreseeable future, assuming you follow a good wash technique.
Alternatively, you could pick up a polisher pack from CCP and go from there.
As for a how to, washing the car how it was demonstrated in the video is almost ideal - you may want to add a pre-soak with snow foam, or follow up with a spray wax but that's the nuts and bolts of it.
The white marks you're seeing after washing the car will be mineral deposits from the tap water. I'm by no means saying don't follow a wash with a QD - I do all the time - all I'm saying is use a good one. Duragloss Fast Clean & Shine is good, Dodo red mist is expensive but good - or you could use a spray wax like Duragloss Aquawax or Zaino Z8 after your wash. But use any of those over an already waxed finish. You don't have to wax every wash (if you use a good wax/sealant) but it certainly doesn't hurt to use a QD after every wash.
Have a flick through these -
How To // Car Care Products - most have videos attached and can be very helpful.