Well, it all depends on what you are making and what you want it to look like.
I have used almost every type and kind of wood that you can imagine. From 3/4" plywood to solid zebrawood. The only thing that affects thickness is the clearance between the table and the upper arm. It seems that about 1-1/2" is the maximum.
As far as kind of wood you can try anything. Personally I do not like pine and cedar because it is too soft and splinters and doesn't make a clean cut. Then on the other end, zebra wood and ebony is just too hard and is hard on blades and slow to cut and is easy to burn.
Most people who do fret cutting use plywood any place between 1/8" and 1/4" thick sometimes 3/8", 1/2" and even 3/4" are used. Just make sure you use the good birch plywood with a lot of ply's and no voids!
Some people who do larger 3D cutting (bowls, vases salt & pepper shakers for example) might use 3/4" plywood or cherry, maple, walnut and almost any exitoc wood available.
So, I guess that means "It's all up to you and what the project is". As you will soon find out, there is "NO right answer" to your question. Just do whatever trips your trigger and do like the rest of us and learn as you go and HAVE FUN!!
Rog