I am a big fan of solfege, you know, "do, re, mi." ("do" is pronounced like "dough", not "doo") There is a lot of (ego-based) disagreement about whether to use the fixed-do or the movable-do system. If you want to develop both the ability to read a melody and, having heard a melody, the ability to write it down, I would practice with movable do. Fixed do is useful for more advanced uses. As far as I can tell, there is no real conflict between the two systems.
I learned solfege using the techniques developed by Zoltan Kodaly, the Hungarian composer, and I recommend them. But I'd advise adults not to worry too much about the hand-signs, they are not the important part, which is the syllables. There are other solfege methods around that are also good. It doesn't really matter which one you use, what you want to do is learn the intervals by ear, so they come to you automatically; and solfege is a great tool for that. Don't just sing scales, that won't get you anywhere. Start singing songs in solfege. For instance, Row, Row, Row Your Boat goes like this:
Do, Do, | Do, Re, Mi | Mi, Re, Mi, Fa | Sol | Do', Do', Do', | Sol, Sol, Sol | Mi, Mi, Mi | Re, Re, Re | Sol, Fa, Mi, Re | Do.
My Bonnie Lies over the ocean starts out with a major 6th, or Sol, Mi:
Sol, | Mi, Re, Do, | Re, Do, La, | Sol, Mi |
(There are supposed to be little marks you can put after each syllable to show what octave you are in, but I can't type them here.)
If you can find a copy of Kodaly's 333 Exersizes, that is a good place to start. They are incredibly boring, but if you do some every day, you will be surprised at how quickly writing and reading music start to become second nature.