If there is not a runner on base. A catcher must make sure they back-up first base on all throws from the infield. Do not stand at home plate and watch. If you want to watch the game, buy a ticket.
Catcher’s must take the appropriate angle to back-up first base. You will need to sprint toward the warning track; get as deep as you can so the ball will not get by you. Run to a spot that the ball will go to if missed by the first baseman. An angle from the third baseman will be further up the line than an angle from the second baseman.
Work hard, and don’t be the reason that a runner advances on a throwing error.
With a potential double play, I have heard two different points of view. I have heard coaches instruct their catchers to remain at home in case the runner at second is safe, and tries to score. I do not subscribe to that philosophy. I instruct my catchers to continue to back-up the first baseman. It is very difficult for a runner to score from first on a ball hit on the infield. If you are worried about the runner, you can assign the third baseman or the pitcher to cover home if need be. I believe a catcher can get back to home in time.
When a ball is hit into the outfield, and there is not a runner on base, the catcher should trail the runner to first base. If the runner takes too wide a turn, the outfielder, or relay man, may throw back to first base. You can either instruct your first baseman to follow the runner off the base (allowing the catcher to cover first on a throw behind), or the catcher can back-up the first baseman on a throw back to first. Either way, get down the line. Hustle will always be rewarded.