There are really many answers to your questions...

1. You can get the wood at a lumber yard, some scrap you have around the house, or you can specialty order online.... slate, if not buying in big bulk check out grassycreekcalls.com. You can also get your sound board through there. Though, like the slate, if buying to make a lot of calls there are better souces.

2. Tools to do the job if making it totally... lathe, chisels, chuck or faceplate. Good chisel sharpener, sand paper in a lot of graduating grits. Goop glue.

3. You can use various finishes. I use a homemade brew I buy from another callmaker. There is poly spar, and MinWax wipe on......

4. Best way to build, or cheapest? Cheapest is to buy all your component pre-turned and cut ready for gluing. Best way is to turn your own ( but it can get expensive depending upon your learning curve). Strikers can be anything from a hand whittled stick (done that and got a turkey doing it) or a dowel fitted into a top piece, which can be a corn cob or another piece of bigger diameter wood. Or you can turn a one piece striker on a lathe.

Hope this helped... I'm no expert, I'm sure someone else can hit the finer points of it.