Electrical – Identifying unknown transformer from a computer PSU

Normally this would only need 4 wires: the common, and taps for the three voltages. Having 6 wires complicates things a bit.
One way to find out is to measure resistance between wires. If there is indeed an common and three taps you should measure \$\dfrac{220}{120}\$ the resistance between 220V and common, compared to the resistance between 120V and common. So look for ratios \$\dfrac{220}{120}\$, \$\dfrac{250}{120}\$ and \$\dfrac{250}{220}\$. If you measure a relatively small resistance between pins, you're probably measuring between 220V and 250V.

Alternatively you can apply a low AC voltage (from the secondary from another transformer) to the secondary and measure voltages between pins, and find out the ratios. Even if you know the secondary's voltage and apply that, don't expect to measure 120V, 220V and 250V, since an unloaded transformer outputs higher than nominal voltage.

Since transformers by their nature are bi-directional, the selection of the primary side totally depends on your input voltage and desired output voltage.

The transformer you describe likely has multiple taps on the "primary" side, may have multiple windings on the "primary" side and likely has multiple windings on the secondary side. Start with a low range DMM, and check for continuity between different leads on each side of the transformer. Once you have mapped continuity, check resistance between the same leads. You should be prepared for the transformer to be as complex as this:

The "secondary" side may be a single coil with multiple taps, or it may have multiple outputs more like the above example.

Once you've reverse-engineered the coil arrangement, you'll need to determine the turns ratio between each set of coils. I would NOT recommend your 120VAC test for this. Start with a much lower (and safer) voltage. Find a small "wall-wart" type power supply that you can sacrifice. The lower the output voltage the better. You want it just for its transformer, not the rectification and regulation components, so if you can find an AC-output wall-wart, you can use it's output as-is. What you want is a low voltage AC source that you can use to test individual windings. Note that applying a low voltage AC source to the "secondary" may result in lethal voltages on the "primary", so be careful!

Find one set of windings to apply your AC input to, and measure the resulting output on each set of coils and on each tap. Transformers are ratiometric, so the relative voltages will be the same using your low voltage AC test vs. when you identify the intended primary winding and apply 115VAC to it.

Doing this, you should have a good sense as to what windings are present that the relative turns ratio between each. Good luck!