Getting to the basic package that comes with a shopsmith is a good start. Since you are just getting started let me say that if you don't have the Power Tool Woodworking for Everyone (PTWFE) book that matches the vintage of your machine that too should be on your list of must haves.
Here is a picture off the site of the tools that come with the shopsmith now, you might have a different model but in general look at items 10, 11, and 12 as lathe items.
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This next image has those items circled so you can get a bit more information on them.
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Once you have these you can do some turning but in most cases the lathe tools are not likely going to be sharp enough to use effectively or at the very least they will not stay that way long. So figure on getting a why to sharpen them or purchase tools with carbide inserts which you replace when needed.
The tail stock uses a "dead" center and a lot of people find that to be a problem and add a "live" center.
Then if you are going to do face plate type turning a face plate or two will be needed. At this point one might look at a lathe chuck with jaws as well.
If you are going to do anything that involves drilling shopsmith sells a piece that mounts in the tail stock and holds the drill chuck so that is a good addition.
If you are going to turn pens then there are more specialty tools for that function. Want to turn bowls then start thinking new tools designed for that purpose.
This is just a short preview of what you might want to start thinking about. If you have an idea of what you want to do that would help identify an order to getting things and which options are open to you.
Ed