While I am by no means a master turner, I’ve been around metal and wood turning lathes a long time. When I started turning wood I bought a set of Harbor Freight carbon steel chisels, and the have served me well. That said, buying sets often leaves you with tools that you never use. I have three ‘go to’ tools that I use for most of my work, a 1” skew that was made for me by a friend, and a 1/2” bowl gouge that I bought from PSI. They are both HSS. I made my own cut off tool from a carbide 1/8 metal cutting cutoff tool. The HF tools were around 20 bucks, and quite useful in learning to sharpen tools. And you will have to sharpen them. You don’t feel as bad when you have to re-grind an inexpensive tool because you mucked up the geometry of the cutting edge. Remember this one point…the lathe is probably your smallest investment…tools and goodies will suck your wallet dry.
Enjoy!
-- One of these hammers oughta fix that...