Okay, that transfer stuff is not going to work in the sunlight. Just take it back to the store and get your money back. The easier way to do this is buy some big paper (the size of your screen) and draw your design onto it one color at a time.
For example, if you're doing a print with red, yellow, and blue: First, draw a "master design" on one piece of paper, exactly like you want it. Color it in with colored pencils, this is your map. Then start with one color, like yellow. Lay another piece of paper over the master, and trace all of the parts that are going to be yellow.
After you've done that, just use an X-acto knife to cut it out and make a stencil of the yellow parts. Be careful about "floaters" (pieces that aren't attached to anything else) you don't want to make any of those if you don't have to. If you do make some, just tape them where they go on the screen and pray that they stay on. After you've made your yellow stencil, tape it to the bottom side of the screen (with your floaters in place). Then just put your paper underneath and do your yellow ink.
Remember to make several prints, not all of them will turn out right. Then let your prints dry and repeat with the other colors you're going to use. Keep in mind, that sometime colors can overlap (if your paint isn't too thick) and can make new colors!
***The easiest thing to do (especially if you've never done it before) is to just do a one color print***
You make stencil out of wax or plastic, cutting out where you want the ink to be. You then apply this to a screen woven of silk. These stencils once were hand made and even twenty years ago required some expensive camera equipment. Today they're cheaper, cheap enough I saw an infomercial advertising a home system last night.
After you have the screen you set it on whatever you're printing on. Silk screening works well with plastics and cloth, although you can do almost any item with it. They're much more versatile than say pad printing. You then use a squeegee to press the ink through the screen onto the item.
Basically, it is just a stencil glued to a screen so you can get crisp lines and have parts that aren't connected to other parts