jsb-
That is a great thing to focus on. Staying aware of more than just the ball, the player with it and his or her defender is going to help you improve and move up.
I suggest talking yourself through things while you watch the ballhandler and defender, "screen coming, keep an eye out for that", those sorts of things. I found talking myself through things really helped. I do that in lots of situations--in fact just Monday night I was working with a newer ref about talking through free throw situations in a scrimmage we were working: Talk yourself through the situation to set it in your mind. For instance, as you are about to administer the free throw, say to yourself: "Blue is shooting, so if white violates it is a held whistle call, if blue violates kill it." I find that really helpful. Just try not to talk outloud
Once you start making off ball calls in your primary you will really start to feel like you are officiating
And the senior guys will notice. If you get a good illegal screen call off-ball, any senior official will see that and realize that means you are improving.
If you played ball, it is the same kind of court awareness from being a player. You are sizing up the D on you but staying aware of your cutting teammate. Same thing. You are watching the ballhandler and defender but thinking to yourself that a screen is likely coming, etc. They say "referee the defense, think like the offense." Part of that is anticipating what the offensive player will do and what the offensive teammates will do--screens, cuts, etc. Dont anticipate calls, but anticipate where the play might go so that you can be ready to make any calls.
Good luck and I hope that helps! Its a good question to ask.