Ben of Throwsticks.com has a good video on the method and technique of making a karli. He doesn't give pointers on what sort of wood to use; but he does name the wood he has in the video. Though really any dense and durable wood works.
https://www.youtube.com/watch?v=zGcg9m6Eabg

I haven't made an airfoil-type throwstick yet, but I expect any type of wood would work, with the durability being the main feature, and the mass that stabilizes it and helps it carry through its flight. Looking at a few incomplete lists of the flora over there, Manna Ash looks like a reliable hardwood, but even spruce would work, and would be easier to learn on as it's softer and hence easier to carve and tune for ideal flight.

As for tools, again I have not made a karli so take my words with a grain of salt: a saw to cut the bit of wood to the desired length, a hatchet for bulk wood removal, a sharp rasp for finer wood removal, and a file to fine tune. And probably a vice to hold the karli while working on it, probably with a rag between the karli and the jaws if the vice is metal, so that it doesn't bite into the wood. These, I expect, are all that's needed. Probably also some sandpaper and wood finish if you want a smoother release and a nicer looking finished karli. Though a drawknife, spokeshave, and a hand plane would really help during the shaping.

It's probably not hard to make an airfoil-type throwstick in all honesty - after all people have made them with stone tools for thousands of years, even up to the modern era. I'll need to experiment as well!

I hope this helps!