Base your screen-printing design on a drawing. It can be as complicated as you like, as long as it is in high contrast. There is nothing wrong with keeping it simple, like a single flower or a geometric shape. Whatever you feel comfortable with, the screen printing process tends to make almost anything look more impressive.
Base your design on a photo. Find a photo of someone or something that you like and trace over it, again making sure that the resulting image is a high contrast version of the photo. This can also be accomplished by scanning the image into Photoshop, changing it to grayscale and then using the highlight/contrast tools to take out the lighter shading that wouldn't translate through screen printing.
When you set up your screen for printing, you must determine what will remain white, and what will be inked. To change up a design you like, invert this selection. This way, if you once had a black outline of a flower, you will now have a black flower on a black background, but the outline will be white instead. This can be done to create visual interest, but it is also nice to have the opposing images framed next to one another for contrast.
If you have a complex image, say of two people standing in front of a landscape, play with alternating dark and light in the foreground and the background. This is similar to the previous section, just taking the idea a little further.
By laying construction paper chunks, leaves or any other flat shapes over your printing paper, you can block out areas where ink may have gone and create a more avant-garde, abstract look. Just print over it and peel the items off. This can add interesting depth, as whatever you printed on the background will peek through the holes made by the shapes. So rather than having to draw or predesign anything, experiment with shape layers.