I got PS+ a couple of days ago so finally got to try out the online racing. Works well, somebody used me as a brake in my first race on the first corner so it wasn't a great start haha, but I won my 2nd race.
How long does it take to rank up SR and DR?
Not that long, it's much easier to go down. ;) The best method to quickly advance (that is if you want to ;)) is to pick daily races without any slow corners at the start of the race (Brands Hatch, Monza) and without walls close to the track (Bathurst, Tokyo). BB Raceway and Kyoto Myabi are great picks imo.