What is the best board (type of wood) for making signs?
I will purchase board and then cut it to size of sign. Need something that doesn't warp.
Since most of what you ave admitted to does not seem to be aimed a "wood" signs, particularly, I would suggest not using any type of wood.
There are durable, exterior-grade, paintable foams made explicitly for the sign-making trade. They can be routed for relief/3d signs but they don't have any tendency to rot or warp.
Of "woods", waterproof Medium Density Overlay (MDO) is probably the most suitable/consistent product. It's a lot heavier than the foams. It has a completely smooth/flat surface (unlike plywood.)
You have a number of choices here, many of which aren't wood-based.
Need something that doesn't warp.
In reality warping is one of the lesser concerns, with swelling and delamination from water intrusion and glue breakdown being more critical factors than the development of a slight bow.
Plywood seems like an obvious recommendation here but it's not as simple as that unfortunately. Plywood quality has been in decline for a long time now and woodworkers regularly complain that what's sold today is noticeably inferior to what was sold just 10 or 15 years ago. And I've seen evidence of this firsthand, with supposedly waterproof plywood (hardwood ply stamped WBP*) delaminating after just a few months of occasional exposure to water.
So if you want a sign that holds up well for a long time suggesting you buy exterior-grade plywood is no longer sufficient. You need to source the best quality exterior ply you can find, and that is probably going to be classed as marine-grade. This stuff is expensive so expect some sticker shock, but it is guaranteed to be waterproof and to withstand extremes of temperature (e.g. from direct strong sunlight).
For a sign that will be painted I think either marine-grade plywood or a waterproof/water-resistant version of MDF are your best options in terms of wood products. Note: the MDF must be sealed fully, that means both faces and all edges, and this is a good idea with any plywood also.
For a sign that features applied vinyl graphics arguably nothing wood-based is the ideal way to go but instead an engineered plastic, metal (e.g. honeycomb aluminium) or composite board material. These are actually made specifically for signage. These boards respond minimally to changes in temperature, not at all to changes in humidity and, in short, few wood products can compete in terms of stability and service life.
The aluminium or aluminium-faced composite boards are also worth considering for painted applications (no wood grain, yay!), but you will need to delve into how to prime aluminium properly prior to paint application. It's not technically any more difficult than priming most other metals but for best results specialised primers should be used.
*Water-and-boil-proof, an older classification for waterproof/water-resistant plywood.