You can read up on these materials, but these are the common ones in the sign industry. MDO, HDU, & PVC. Each have their pros and cons. MDO for example cannot be carved and will eventually fail from water seeping into the ply. Sealing the edges will help but it's still an inevitability. HDU is great for carving but takes a bit of work to finish and can be tricky to mount. PVC takes longer to carve and can be on the heavier side but is very durable.
If you're doing a painted, vinyl, or letter applied sign MDO is probably the best bet. Anything carved you're going to be looking at PVC or HDU. Also don't be afraid to fake thickness. Buy a piece of 3/4" material and then build up the sides to make the panel look thicker, similar to how they do counter tops.