Regardless which side of the plate you hit from, your first step out of the box is a crossover step. Drive out of the box as if you were stealing a base. Get in the habit of getting out of the box quickly; this helps you at the close play at first and may just turn that long single into a double.
As you take your first couple steps, pick up the ball. Assume there will be a play at first base on any ball hit on the ground or on a line. Once you know where the ball is heading, focus in on where you are going.
Few players in the Majors run out every ball. As with many things in the big leagues, it may okay to do it when you get there but you shouldn't do it when you're working to get there. Lead by example, run every ball out and run it out hard. It doesn't matter if it's a slow roller to the pitcher or a pop fly to the center fielder, get your legs going and hustle down the line.
Once you've determined that the ball is going to be fielded by an infielder, focus on the front part of first base. As you hit the base, start breaking down. This gives the illusion to the umpire that you were at the bag sooner than if you breakdown after you pass the base. After you have crossed the base, slow down and look to your right to see if there was an errant throw.
If the ball is going to make it past the infield, get ready to round first base. Do this by moving off the line and into foul territory to set yourself up for the turn at first. Try to hit the inside corner of the bag. Be aggressive with your turn, put pressure on the outfield to field the ball cleanly and make a good throw. There's a fine line between being aggressive and being stupid. Put yourself in position to take advantage of any mistakes but don't get caught being too far off the base.
Hustle is mandatory! As a team rule, every player must run hard on all hit balls. Make sure there is a penalty for breaking the rule (i.e. - Player will not start the next game.) and stick to it. Stand by the penalty, no exceptions.