The Bowyer's Bible series has a few plans in them. It's a good reference. Also, I still reference Sam Harper's site (poor folk bows). Lots of good information there as well as detailed plans.
Your most common backings will be hickory, bamboo, and maple. All should be about 1/8" thick and have grain as straight as possible (bamboo always has straight grain).
As far as belly thickness, you would have to choose a material and then we can help you out. If you find plans you like you can copy them for similar SG materials. I group most of the white woods together, hickory, oak, elm, maple. And they normally start out around .75" thick at the fades with a taper of around 3/8". High specific gravity woods you can get by with less thickness and less taper. These are just rough ideas, but let us know what wood combo you want to try depending on what's available to you and we can point you in the right direction.
I wouldn't try a tri-lam at this point. You can make a really nice backed bow without as much investment (both time and money). But if you want to try we can point you in the right direction there too.