I think somebody on here mentioned a midi lathe. That's what I'd recommend as well as a first lathe. You can make pens, handles, smaller bowls, etc, and if you add a bed extension later you can do table legs and the like. I've got an older Delta midi that I like for small and medium size stuff, JET makes a good one, or go to the Grizzly website and look around.
I wouldn't recommend a Shopsmith if you're mostly interested in turning. You want a table saw, drill press, etc. fine, but realize there's some limitations.
If you're just starting out, I would buy a set of 3 carbide tipped chisels. I have probably 30+ chisels, probably at least 2K in them, and I find myself reaching for my midsize carbide rougher or finisher as often as not, unless there's a specialized cut I'm doing. Conventional chisels have to be sharpened a lot, don't come sharp, and it is a big learning curve if you're on your own to learn how to do that, and it takes a fair investment in sharpening tools or system. The carbide tools are not sharped - use them till dull, then switch around the carbide tip, then after a couple of years bolt on a new tip. Super, super simple. And the cutting technique is different, and in my mind safer than with conventional chisels. Here's what I'd do starting out:
Lathe - here's a JET, there are others around, this is about what to expect to pay. Should be a cast iron so you have enough weight to keep it steady and balanced. Please don't go to harbor freight for this. Look at craig's list
https://www.woodcraft.com/products/jet-variable-speed-mini-lathe-model-1015vs
tools
I like this brand, and this size for the midi lathe. You can use these for everything from bowls, pens, legs, etc.
https://www.woodcraft.com/products/...1e3c69702d6593002bd2,57641e5169702d2f2f0001ba
that's the finisher. That and a rougher will get you going
The lathe will come with a tailstock and headstock, so you can turn sticks. Should come with a basic faceplate for bowls as well. Once you get hooked, you'll add all kinds of dodads to your system.
Turning is my favorite thing to do in the shop, and I do a lot of cabinet work. It's a ton of fun and you can really zone out on the lathe. Very relaxing and satisfying, doesn't make much noise, and you don't need much space to set up a nice area to work.