Honestly, I started with BF3 back in the day. A lot of people on PC in FPS games are hella good, so you can get really good just playing against them.
I'd start with some BF4 and play that along with CS:GO. it'll take a while to get really good, but you'll have fun doing it.
Also play on a low-ish sensitivity. I don't know what DPI you play at now, but from experience (I've played both really high DPI and really low DPI) I'd play at like 800-1000 DPI to be as accurate as you can. This is a personal preference, but from what I've seen out of all of the best players around the world and the good players I've played with, low DPI is the best choice.