I did the same on the same dark blue paint using my old wash brush that I used on every car because I didn't know better on the soft paint on this car. Here are the steps I took to get rid of scratches. Didn't get it all out but got rid of majority to presentable and acceptable
1. clay bar the car (you can probably skip this step since your car is brand new)
2. Went with Griot's polish system because I had friends that had show cars and use it often. Polish is very very mild and the applicator is designed to prevent amateur to mess up the paint (polish pads basically stops spinning if too much pressure). Polish basically smooth out the hard edges on scratch and/or remove it complete it from the top coat.
3. Use a filler after polish. Filler fills in the scratches and hides the sharp edges from the light
4. Apply wax
You can also just take it to a detailer to do #1 followed by applying something like opti-coat. They will probably take a few hours to remove the scratches at $75?/hour.
As for washing, after seeing how easy this car is to scratch... I decided to follow this detailer's method of washing a car. I followed this video
Tesla Opticoat Pro Treatment - YouTube
I learned the best way to wash cars is in multiple passes. Each pass remove bigger to smaller stuff. So I invested in a electric power washer + foam gun. Makes it go pretty fast.
step1: power wash to spray down and get rid of big dirt
step2: foam the car, gets rid of more dirt
step3: power wash rinse
step4: hand wash, 2 bucket system. If you use microfiber towels, be sure to remove the tag as they scratch. Best to actually use a microfiber wash mitten which has more depth to hide the dirt so they don't stay on the wash surface to scratch the paint more. By the time you get to this step, whats on the car is mostly just a film of dirt left on the car as prior steps already removed the big dirt that might get caught in your towel and scratch the paint. Wash from top of the car to bottom (where most of the bigger dirt is)
step5: use a water blade to get rid of most of the water, microfiber towels to pat the rest off
The best quote I heard from a local detailer is... you will be a slave to the car
He suggested to detail it 2X/year (spring, fall) where I live (rainy Seattle)