I am not a musical person by occupation but more by vocation and heart 
; meaning the less theoretical the approaches would appeal most to mm, while I do understand that it will take a combination of both. 
Stephen,
Learning to play by ear is for me forst and foremost a practical thing!
Most of us - probably you, too - started learning songs by ear as soon as we could talk. Nursery rhymes, Christmas carols, that sort of thing. We did this with no knowledge of music theory and without the ability to read (either tune or lyrics!). The method we instinctively used was listening frequently, then trying to sing along, and then trying to sing them by ourselves. Our first attempts were a bit off tune, but as our voices developed, and we got more accustomed to using them, our pitch improved. That is, we learned how to work our vocal cords to get the precise interval to the next note. We could sing the first 4 notes of "Baa,baa, black sheep" long before we learned that the interval between note 2 and 3 is a fifth. Having the tune in our heads allowed us to check whether we were doing it right.
With a new instrument - I just recently star5ed learning teh Crane Duet - I just add a step to that. Listen often, hum along until I can do it confidently by myself - and then take the instrument to hand!
What then happens is what small children often do when singing - I don't hit the correct next note! But, having the tune in my ear, I notice whether I'm too high or too low, and can find the right note and correct it next time round. I just keep doing this until I can play the tune through without hesitation.
Then I take the next new tune, and find that it's easier than the first! This is because some of the jumps between notes are the same as in the first tune, and you know how to play them. And the more tunes you try, the more situations you will recognise when you start a new one. Practice builds experience, and experience is perhaps the most important aspect of playing by ear. When you've learnt, say, 20 tunes from a particular genre, a new tune from the same genre presents hardly any challenges that you haven't faced before.
The goal is to be able to take a tune that you've got in your head (i.e. that you can sing, hum or whistle) and play it through for te first time on the instrument with only a few mistakes first tim round, and only one or two mistakes second time round (there will probably always be a couple of places that remain "difficult" for quite some time!). This goal is by no means unrealistic - it just takes daily practice for as long as it takes.
Of course, as an adult, you can make use of theory to underpin your practice. For example, it's easier to find the right next note if you exclude all the potential wrong notes, for a start. That means learning scales. If you play the tune in G major, then you will only need the notes of the G-major scale, and the next note must be one of them. It's also good to know that there are tunes that start on the tonic note (e.g. G in the key of G), and others that start on the note a 4th below (e.g. D in the key of G).
Another essential trick is to always start an attempt at playing by ear by playing the tonic chord of the key you want to play it in. Playing by ear is, as the term implies, guided by your ear. And your ear needs a point of reference to work from when judging where to go from where you are, and whether you've done it right.
You being an EC player, harmonising by ear is probably not as important as it is to me as a Duet learner - but here, too, the principle is the same: listen often, chord along until you can follow the lead, then play the chords by youself while you or someone else plays the tune. The more tunes you learn this way, the easier it gets.
In short - just do it! It's a very practical process, but it takes time and frequent practice. Little and often is better than much and seldom.
As to breaking the tune down into phrases and learning each phrase separately: I wouldn't do that when starting to learn a new tune by ear. I'd try to play the tune through as well as I can. The breaking down into phrases helps when you've already found out what notes you need, and are trying to get your fingers to play them in the right sequence at the right tempo. But that's not about playing by ear - it's about playing from memory, and applies to tunes you learn from the dots, too!
Hope this helps,
Cheers,
John
