Print out the design you'd like to use and cut it out, then lay the pieces down on the sweatshirt where you want them.
Carefully trace around each letter.
Turn the sweatshirt inside out and lay a piece of the contrast fabric with the right side (printed side) facing down to the sweatshirt. Pin it along the edges. Flip sweatshirt right side out and sandwich both layers of fabric between an embroidery hoop. (If you don't use a hoop, use lots of pins to ensure the fabric doesn't move around.)
Thread the needle with embroidery thread (using all six strands as one) and tie a knot in one end. Begin embroidering along the lines you have traced. Poke the needle up from inside and pull the thread through to the knot. Moving along the line, poke the needle back down 1/4 inch away from where you came up and pull the thread through.
Moving along the line again, poke the needle up 1/4 inch away from where you just went down. Now you'll go BACK, poking the needle down at the end of the stitch you just made.
Continue to backstitch until you have completed the outline of the letter. Come up 1/4 inch past the end of your last stitch, then go back and poke the needle down at the end of your last stitch. This gives you a continuous stitch line. When you need to start new thread, knot the thread inside the sweatshirt.
Continue until all letters are outlined.
Turn the sweatshirt inside out and use pinking shears to trim away excess fabric, being sure to leave an inch border of the contrast fabric (that is more than you see in the photo here - if you trim too closely when you wash the shirt the fabric may fray past the embroidery).
Turn the sweatshirt right side out and pull the two layers of fabric apart inside one of the letters. Using fabric scissors, carefully snip just the sweatshirt layer to make a small hole. Then cut just the fabric material inside the embroidered lines. Take care not to cut through any of the embroidery, and not to cut through the contrast fabric at any point. Just use small snips to cut away only the sweatshirt fabric.
Repeat the process with each letter until your entire design is revealed.