Seems to me there are three fronts being advanced in this thread:
Learning songs, learning chords, and learning tunes.

They're all slightly different, but all depend on listening a good deal.

Songs (vocalizing words to a melody) are probably simplest because words, as you memorize them, have the tune notes sort of embedded in them. They are their own built in mnemonics. And about as soon as you learn all the words, you already know the tune. I read somewhere that some cultures can't conceive of a melody without words.

Chords are a different trick. After you've learned the words try playing along with your voice on a chordal instrument. You will probably hear where the voice tune and the chords on the instrument don't match, or the chords want to pull your voice away from the melody you know. That's a pretty good sign you should be changing chords. Martin Gibson's comment above about listening to the "drone pitch" is a pretty accurate way of describing this.

Tunes are perhaps the hardest to learn by ear and since you play fiddle, you're on one of the harder instruments to start out on. You can't fall back on chords like on a guitar, mandolin or banjo.
You need to know your scales on the fiddle pretty well before you start. And then playing a lot with other people helps.

A fine guitar player told me that when he sets out to learn a new tune by ear, he gets a recording of it and listens to it at least 10 times in a row so that he can hum the melody before he even picks up his instrument. Then he figures out what key it's in, and then starts trying to find the notes, matching them against the tune he's humming in this head.

As a fiddle player you should know about "mouth music" or "deedleing"
which is the way the old timers memorized and, at times, transmitted tunes without having an instrument. It's like humming, but instead vocalizing the tune with "deedle deedle dees" - I picked up a CD several years ago of mouth music in various cultures, Celtic to Appalachian, and it was pretty informative. There's also been some discussion here at Mudcat about mouth music.

So - listen to a tune a lot, pick up your instrument and try to find the first few notes, then go back and repeat the process. It'l get faster each time.