The older Craftsman lathes would be a good starter wood lathe.
That's what I had a few years ago. When it was taking up too much room at the shop and I didn't have room at home, I got rid of it. Mine was a 4' Craftsman with Craftsman turning tools, which were quite nice. I've seen them on Craigslist for around $200. Now that I'm retired, and have room, I may just get another one.
I mostly used mine for making custom gavels to sell at the shop, and for making duplicates for my dad, when he was restoring antique furniture. My competitor at the shop was a retired shop teacher, so he showed me how to use it. They make special stones to sharpen the gouges, but I just used a whetstone to keep them touched up.
I still have a stash of exotic woods, like Brazilian Rosewood and Osage Orange, and I have a bunch of black walnut.