I do a few turned ones. There are some things I would look for in a lathe.
First off you're going to want 36" or more between centers. This is going to rule out a lot of the mini lathes unless they have a bed extension. If you want to sell canes I've found it best to make them long and cut them to size when they sell. I use my daughter's lathe that has about 30" and a home made bed extension. It's not ideal.
Look for a lathe that has a threaded headstock with a morse taper inside it. #2 Morse taper is easiest to find accessories for, older Craftsman lathes used a #1 Morse taper. Accessories are still available but you have to look a little harder. Unless it's a really obscure thread size on the headstock most chuck manufacturers will make an adapter to fit it.
A lathe with a cast iron bed would be my first choice.
You want a tailstock with a Morse taper as well for the same reason.
A lot of cheap Asian lathes had square tube beds and a non-removable centers. Avoid those like the plague. They're wasted time and money.
While you're waiting to buy a lathe you can accomplish a lot with a spoke shave or even a good sharp hand plane.