For beginners all you need is plasticine clay (i.e., nondrying oil based clay, as made by Crayola, etc.). All-purpose craft stores have neutral and natural colored plasticine clays.
For more advanced work, one can use clays that dry. For permanent projects I use regular potter's or modeling clay that is fireable. This can also be bought at craft stores.
You'll find that this type of clay is more susceptible to nuances in stylus pressure. Because this clay dries, you can also let it dry out a bit to write smaller cuneiform signs (dryer clay allows for writing shallower and hence smaller signs). This type of clay also shrinks when it dries and thus creates artful contours in the signs (see the image here as well as here). You can also experiment with big projects in this type of clay. This type of clay allows you to preserve your work.
While potter's clay is fireable, it is not necessary to fire any of the tablets that you make using it. It air dries hard enough to last, and is durable enough if the tablets don't get knocked around excessively (and if your dog doesn't eat them--really, this happened to me; the dog did eat my homework). In antiquity, the majority of everyday tablets were not fired.
Further reading:
On the modern firing of cuneiform clay tablets for preservation, see:
Thicket, David; Marianne Odlyha; and Denise Ling. "An Improved Firing Treatment for Cuneiform Tablets." Studies in Conservation 47/1 (2002): 1-11.
