There are two ways to screen print with a squeegee, either push or pull the squeegee. Pulling the squeegee toward you is the traditional method, but I usually push the squeegee when manually screen printing for long periods of time.
Before you push or pull your squeegee to print you can flood the screen with ink. Flooding the screen is just "loading" the mesh. You are not pushing the ink all the way through the screen when flooding. Your moving ink into the mesh to set up your print stroke. Once you have flooded the mesh- you can do your print stroke. Either pushing or pulling :) Pushing or pulling is almost a Chevy or Ford type of argument- there is people who passionately argue one is better then the other. My view is- as long as the customer is happy with the end result then choose your print stroke. Sometimes I pull on a manual press and sometimes I push on a manual press.
Either pushing or pulling you should only be applying enough pressure when printing plastisol to clear the screen. When you apply to much pressure with plastisol ink it pushes the ink down into the fabric, and this is not what you want to do with plastisol. This will result in dull colors. Plastisol is designed to sit on top of the fabric. So only print with enough force to clear the mesh and leave the ink deposited on top of the garments. This will keep the colors bright and help achieve a more opaque print.
Whichever method you choose to use, pushing or pulling, just print the shirt in the same direction. Never do a push stroke then do a pull stroke. By doing both a push and pull stroke you run the risk of a blurry and smeared print. However, there are exceptions to rules- I have done both a push and pull stroke when the registration was a little off on the last few prints- and by doing a print in each direction with the squeegee i was able to cover the base with the top colors.
With water base or discharge ink you should be applying more pressure. Water base and discharge inks are pushed into the fabric. So use less off contact and more pressure when printing with water base or discharge ink. Push the ink down into the fabric.