You want to play deep enough that you can cover as much ground as possible, while being shallow enough to make it to first base prior to a throw being made from a position player. It would be difficult for a shortstop for example to have to make a throw to first base if you are still running there while he is delivering the ball. You need to get in position to give the position player a target to throw to.
Even though we want to be able to depend on the pitcher to cover first base on a ball you field to your right, we don't want to have to depend on him for every ball hit your way. Young pitchers will often forget to cover the base. Make sure you are shallow enough that you can catch the routine ground ball and tag the base yourself.
Right-handed first baseman: Position your right foot against the bag with both heels about parallel to the base line. You want to be in an athletic position with your knees bent.
Left-handed first baseman: Position your right foot against the bag. Since you will be sweeping a tag with your right arm, you can open up more toward the pitcher.
As the pitcher goes into his stretch give him a target.
If he goes to the plate shuffle a few steps toward second base and get ready to field any ball hit your way.
The following is a good video on some important aspects of playing first base by Don Mattingly.
When catching the ball from infielders, you need to get to the bag as quickly as possible. Once there, turn and face the direction the ball is coming from. Make sure you do not stand on the bag in a position that doesn't allow the runner room to run by. Place your throwing hand foot on the edge of the base. Make sure you are in an athletic stance ready to move (don't assume the ball will be thrown right to you). You shouldn't stretch for the baseball until it is well on the way.
The image below shows a first baseman waiting for a throw from second and from third. No stretch will be made until the ball is on the way.
Two common mistakes are made by many players:
1. Stretching towards the ball too soon.
This puts you in a position of no return. Say for example a throw is made directly at you. You stretch out to catch the ball and realize that the ball is going to hit the dirt and it's not going to be an easy short hop. Your stuck. There is no adjustment that you can make other than trying to knock it down the best you can. If you would have waited, you would have options on how to field the ball.
2. Trying to stretch as far as you can on every play.
On a bang bang play, stretching for the ball can be the difference between out and safe. On balls thrown off target it is necessary in order to keep your foot on the base. On plays that aren't close and the ball is thrown at you, stretching out only adds a level of difficulty that's not necessary to get the out.
If the shortstop throws the ball directly at you or to your glove hand side, step out with your left foot (for a right-handed first baseman) in the direction of where you want to catch the ball. So, if the ball was thrown a few feet on the home plate side of first, you may need to step towards home (not directly down the line) to make the catch.
On a ball thrown to your throwing hand side, step across your body and catch the ball backhanded.
The image to the below shows an example of stepping towards each type of ball described.
Work on balls being thrown to the side of the first baseman. Make sure he does not step in the baseline on either side of the bag. It's a sure way for the runner to end up colliding with him. It's a dangerous situation for both players, so make sure all your first basemen know where to step to receive the ball.