I have used clay both from Eastwood and JC Whitney. I'll remember in the future that AutoZone also has it.
I have a few suggestions for you based on my own experiences. First, wash the car completely like you normally would. Get all the big bits of dirt, bugs, tar, etc. off the car. Wash under the wheel wells so that no dirt is dripping off the car from anywhere. Start with a clean car and work in a clean place.
Once you've washed the car, move it to a place where you can work for a couple of hours, preferably in the shade. Before you park the car there, place a very large clean tarp or plastic sheet on the ground, then park on the clean sheet and remove your shoes while working on the car. (More on this later). For reference run the BACK of your hand all over the car. Remember how it feels, remember the slightly rough texture you'll find in various spots.
Fold the clay multiple times to soften and warm it. Finally flatten the clay out into a disk that you can hold against your flattened palm. Use a lube or water with a little bit of dish detergent in it to wet a small area perhaps 2 or 3 square feet. Use the clay in small circles just like you were hand polishing the car. Just like washing, work from the top down in small areas until you've done the whole car. Break the task into sections so you don't forget any area and to keep the magnitude of the task under control. Keep the area damp with water or the lube. It's very easy to get the warm clay stuck to the paint. Before moving to the next area, use the back of your hand to re-inspect the surface. The area you've cleaned should be significantly smoother to the touch. When completely done, re-wash the car to remove any lube or soap.
IF you did not put the plastic down be EXTREMELY careful to NOT drop the clay. If you drop it on the driveway or other source of dirt the clay will pick up ANYTHING it lands on. It will turn the clay into sandpaper. Yes, you can fold the clay over to put the dirt in the middle but eventually after multiple foldings the dirt/grit will resurface.
Clay is a great way to remove contaminants that would otherwise require lots of rubbing compound or chemicals. However, use it to remove the small stuff not easily seen. Clean the car thoroughly before using the clay. As mentioned above, other products can be used to restore the gloss and protect the cleaned surface. The final results after washing, clay, light compounding (or glazing), and waxing will be a significant improvement.