How do I figure out the windings of a transformer
Perhaps the simplest way is to wire the transformer up and test the whole unit, with a current limiting load in series with a primary. A mains filament light bulb would be suitable (make sure it's filament, not LED or CCFL). Or a mid power resistive load like curling tongs. Whatever load you choose, it should not draw more current than your transformer input is fused for. This does mean you'd have to connect things mains side. If you're not happy doing that safely, then you shouldn't be inside with the transformer mains connections anyway!

Wired correctly, the transformer will produce a significant output, and the bulb will be dim or off.

Wired incorrectly, the transformer will present a short circuit to the input. It will produce no output, and the bulb will light at more or less full brightness.