Want to add sub-panel in my barn - 115v, 60amp?
If you have a main panel at the barn that would support a 60A feed, you're very likely in luck - because that would be a 240V panel, (you're thinking 230 but you're outdated) and all you need for 60A worth of 120V circuits is a 30A 240V 4-wire feed (Hot, Hot, Neutral and Ground.)
Were it my barn I didn't want to burn down, I'd run EMT (electrical metallic tube, the lightest form of conduit) because barns have rodents, but if I was not into  learning how to run conduit (or hire that part of the job out) I'd run 10/3 MC cable with a flexible metallic armor to resist rodent teeth.
I would use 20A (and 12 Ga) for the tool circuits, 15A (and 14 Ga) for the lighting circuit(s) unless planning to go terribly retro and inefficient for the lights, and recall that subpanel size (and what the breakers add up to) does not need to match the feed breaker (It's perfectly fine to put a 100A subpanel and feed it with a 30A feed breaker.) You'll only run into &quot;size&quot; problems if you pull more than 30A from one &quot;leg&quot; or hot-to-neutral (for 120V) at one time.
You may also be able to rewire the lathe motor to run on 240V (at 7.5A) and install a 15A 240V outlet (with 14Ga wire) for that purpose. Many such motors can be rewired that way.