I have an old Delta lathe that is full size. I, like a few of the others, would recommend buying used. Personally, I wouldn't go mini, as it limits you to what you can do size wise. Variable speed is nice, but isn't needed.
A good set of chisels is needed. Also a bench grinder to keep the tools sharp. I sharpen mine to a mirror finish by hand and keep a leather strop nearby to keep the edge. Keeping the tools sharp is a MUST. It is not difficult to do, but is time consuming.
Also for safety sake, a full face shield would be prudent to have. I have one and it has saved my ugly mug from getting even uglier. There are two turners in this world. The ones who have not had their work "blow up" on them, and those that will. Be careful!
Wood turning is a fun medium. It truly is amazing what you can do with a lathe.