I think M.Ted has it right. I would simplify it. (It helps to know your scales but you can also count on your fingers.)

1. Take a chord progression from a song.
2. Turn the chord (relationships) into numbers (preferably Roman Numerals)
3. Turn the "numbers" into another key.
4. Eventually train the ear to hear "numbers"

1V //// | I////| V7 ////| I //// ||

This progression works for This Land is Your Land, Midnight Special, Little Darlin' Pal of Mine, many Calypso chord patterns, in 3/4 time, Deportees, Jolie Blonde and probably many more.

Dealing with minor keys is a little more tricky.

Parallel minor keys are just like the major keys but labeled minor such as
IV min //// | I min //// | V7 //// | I min //// ||

The tricky part is when you deal with Relative Minor Keys. This is always the VI minor of any given key. (It has the same key signature as the "Relative Major".

A minor has the same key signature as C major. A minor is the VI minor of C.

But you can sort out that mess when applied. Sometimes you have to have dual labels for chords.

For example: Ghost Riders in the Sky

VI min //// | I //// | VI min //// | ////| etc. or it could be written

I min //// | bIII (major) //// | I min //// | ////

Where did that bIII (Flat III) come from? Well this is where it gets a little more advanced.
Here you might have to know your scales to make it work.

If Ghost Riders is in C major, no problem. Most would say that it is.
but if it was in A minor, which it possibly might be thought of that way,
the III in A is C#. C (natural) would be a Flat III in A.

In short, stick with the easier tunes....I, IV and V to start.

But this is the most important thing....you've got to those numbers to transpose quickly. You do that by practicing changing the chords to numbers as many new keys as you can and them. Eventually, you will hear the numbers.

Frank Hamilton