I think a good used lathe is definitely the way to go if possible. I have purchased 3 of them. The second one came with a lot of turning tools, chucks and accessories. At $800, it was about 30% of what everything would have cost if new. Until a year and a half ago, I thought it had about everything I might ever want to do. Some of the extra’s I still haven’t gotten around to trying out.
My third used lathe with accessories was $1500. This time about 35% of what everything would have cost if new. If I added up all the other stuff purchased as well over the last year, those numbers would easily double. Good luck with the used. You won't spend much that way!