What you need to do is use a paint cleaner or even regular dishwashing liquid to remove all the old wax from your cars paint. After that mix up a bucket of some nice foamy car wash, or use the bottle of detailing liquid that came with the clay. Squeeze some of the foam and water onto your cars paint and rub the clay back and forth NOT in a circular motion over your cars paint.
Many people suggest rolling the clay every now and then so you don't drag some of the dirt it has picked out of the paint over the rest of your car. Whilst that might MINIMIZE the scratches from the ingrained dirt particles it won't eliminate them. The best is to cut the clay bar into 8 equally sized pieces, after you have cut them roll them and flatten them into coin sized pieces. Use one for 1/2 the roof, 1 for rear hatch, 1 for one sides doors, 1 for the bonnet and then 1 for the front and 1 for the rear bumper. By doing this you are always using a clean piece without risking dragging dirt across your entire cars paint causing scratches.
Be carefull to always make sure the cars paint is lubricated with either the soapy solution or the detailing spray. At first the clay will feel as if it is grabbing at your paint, after a few passes it will slide like soap across glass. The grabbing is actually the clay pulling all the ingrained dirt out of your cars paint, once it has pulled all of it out it will be smooth as glass.
Just work slowly and always keep the paint wet and slippery and you will be fine. I usually cup my hand under the clay, that way I keep a small amount of soap in constant contact with the paint and clay pieces.