I think one of the most common problems in any academic subject is that we tend to over intellectualize the process of learning at onset. In my experience, both from learning to sight sing/read and teaching, is that it should be paired with ear training. Through my experience in the professional world both are used simultaneously throughout orchestral and choral settings. I also believe any relevance the learning has to our everyday lives only speeds the process of understanding; even my former elementary aged students progressed in their ear training once I tied it to what they hear in their environment. Improvisation is a skill I would only practice after mastering sight singing/reading and ear training. It relies on the foundation of these two skills so that a musician can hear/play their melodies and harmonies in the context of what is actually prescribed in the music. This doesn't mean 'never improvise' without the skills, but simply that it shouldn't be a primary area of focus; at least in my opinion. It can be an exercise to increase awareness and can be given parameters.
The quickest way to ear train is to take small steps and work to understand to the most popularly used scales and the progression that each scales requires. From there, interval based relationships can be extracted not necessarily based on the name of each note such as 'C' to 'E' or 'Eb' to 'Bb', but from 'Do' to 'Mi' or 'Fa' to 'Do'. Any recognition of note name early on only complicates the process and lends itself more to theory rather than ear training and sight reading. It is my belief that sight singing improves once the prior is mastered. Student who struggle with these skills must learn to not complicate the process and understand their environment. Our world is full of noise and sound. Silence is a rarity and the aforementioned skills can be practiced anytime. Television, radio ads, and mobile phones all provide moments for our skills to be practiced and mastered. Personally, I used church hymnals to practice my sight singing constantly. The variation of melodies really helped me solidify my skills so that I also hear what I've practiced, in other mediums such as the radio and tv.
With all education, there must be a connection in order to solidify and understanding, and just like any other subject, when we first learn, it is the connection to that subject that pulls us closer to understanding. It is fine to simply say 2 + 2 = 4, but isn't it more meaningful to then count it with money or objects. Mathematics in the modern classroom relies on manipulative so students interact with math outside of pencil and paper. Ear training and sight singing/reading should function in the same manner. When understanding a major scale, you should look and listen for examples in music that employ a major scale, exclusively and if it doesn't fit, move it to the side. Create a small focus and then expand the focus as comprehension is increased. Major and Minor Scales (Natural, Harmonic, Melodic) should be the primary tunnel of focus and will take care of the vast majority of popular music that we hear on the radio. The rest of the modes should matter most after such mastery as they will entail small variations on the prior mentioned scales and make your life easier in terms of recognition.