Making springs is not as much fun as you might think but it's a good skill to add ... I've always purchased springs but every once in a while, a special requirement comes up.
I'm no expert, so take this with a grain of salt, but here's the way I made twenty tapered compression springs the last time around.
Made a mild steel arbor, turned to about 2/3 of the desired spring I.D. At the headstock end, I drilled a hole to insert the music wire and hold it. (The arbor was center drilled and supported by the tailstock center.) After the spring is wound, the "tail" is nipped off.
Then I made a wire guide from brass. The wire guide (about 3/4 x 1/2 x 4) fit in the lathe toolpost and had a hole drilled at an angle to feed the music wire through. The guide was slotted THROUGH the guide hole and about an inch past it. A 10-32 shcs was located about 1/2 inch back from the guide hole and was used to provide friction on the music wire, like the clamping arrangement on a pinch knurl.
The lathe was in back gear, about 20 RPM, set up to feed from the headstock toward the tailstock. The lathe gearbox is set for something less than the desired spring pitch (this has to be determined experimentally).
To wind the spring, I'd move the lathe carriage until the brass wire guide was lined up with the hole on the arbor. Then loosen the friction screw on the brass guide and feed the music wire through the guide hole and through the hole in the arbor.
Tighten the friction screw, engage the lead screw (rotate spindle by hand to engage and take up the lash) and turn on the lathe. Turn off the lathe about one or two turns BEFORE the spring is complete. Depending on your machine's characteristics, it will coast for a bit.
I found that the spring-back of the music wire was always more than my estimates and that the mandrels needed to be very small in order to account for the spring-back.
It's a tedious and time-consuming operation. Not something one wants to do routinely if commercially available springs can be obtained. BUT, as I said, I'm a novice spring-winder and there are probably way better methods to do this than mine.
Have fun !