I think first, the best thing you can do for yourself, is try to first brake any bad movement habbits you have, by just focusing on not doing them, think of what kind of movements you wanna be able to do, and practice them, until you can simply think of what you wanna do, then do it. Vs other players, or just running around in an empty map. Is sounds silly, but you wanna know you even can do something, before you try to apply it to an actual situation.
Then when you actually do play people, start trying to apply the mentality of changing how you move to your opponents weapon. That in itself, as we all say it, makes sense, and seems easy, but I'd imagine it really is the hardest part. I personally don't remember how I got any good at simply identifying a weapon in an opponents hand, I guess I just learned it the hard way. Sometimes, it can make a huge difference if you know before they fire it, seeing it fired at you being the easiest way.
Then naturally, you try to apply these moves.
Most times people will lead with hit-scan, so the best way to start is: See your opponent, and don't dodge. Strafe about irradically as they fire the scans at you. If they change out and throw a projectile, you get to test your reflexes.