How to learn the sounds in a language (English)

I believe this is an extremely important question. The answer, if a good one exists, may give some people "superpowers". This applies to all languages a person wants to study, but here we will discuss mainly English, since it's the one I'm most interested in.
A thing to note first: I know there are probably some people that are not linguistically inclined, some that are not motivated, etc., but here we will only consider learners who don't seem to have trouble learning languages.

The question in brief: how can a learner recognize the features in a spoken language, so that they can identify the basic "bricks" (consonant set, vowel set, assimilation, variations) and then try to use them in their speech?

The question in detail:
Until now, my personal theory has been "you can't speak English and understand it easily unless you learn the basic sounds". Like children: first they try to get the sounds, then they try the syllables, and they they can put them together... mom-ma... dad-dy...
My native language doesn't have the vowel in "ship". The result is every learner here pronounce "ship" and "sheep" the same, as "sheep". I had to learn the difference, so now I know that there are two different vowels, used for different syllables.
The point is: I had to learn about that, from books. But that's limiting, and that's the problem. Books won't tell you everything. American Accent Training deals with a lot of features of American English, but for example it doesn't even mention anything about aspiration of T, K, and P, so I presume that's left for the student to notice by themselves. Another example could be transcriptions in dictionaries or any other learning material: the are not precise, so you can't understand any features from there (example: "can" and "cat" are always transcribed as having the same vowel, yet it is different for most Americans).
I hope the question is clearer now: how do I learn those features by myself if no one teaches them to me? How do I actually "hear" them, how do I notice?

I have read here about "shadowing", but I didn't understand how it's supposed to work. If I try to repeat while another one is talking, I won't hear what they say anymore. Repeating little pieces alternating like "me-audiofile-me-audiofile-me-audiofile" might be easier...

The only technique I have tried so far is "attentive listening", similar to "attentive reading" proposed by Tom here. I pick a video on Youtube, and I don't pay attenchin to what they say, but how they say it. Attenchin. To what they sigh. Say. :-) The problem with my technique is that often you can't tell what sound has been used (especially consonants in clusters or assimilated), especially in fast speech. The fact that children don't learn to speak by listening to fast speech leads me to think slow speech is absolutely required.

I'll appreciate your advice on this.

Johnny, if when shadowing, you cannot hear the target speaker while speaking yourself, the volume on the recorder needs to be increased.

Ideally, your spoken volume should match the volume of the target speaker; if the target speaker is too loud, you won't be able to hear yourself speak, and vice versa.

To Jasper

I think Johnny is talking about watching a movie and repeating what a character is saying. So he couldn't catch up w/ what comes next.

To Johnny

The 'a's in 'can' and 'cat' are different? I never noticed that. They sound the same to me.

Yes, can't repeat right away, so you need to "study" the sentences first. And when speak over the audio track you are going to "disturb" it, it's inevitable. Paying attention to an audio track and speaking at the same time is confusing... the feedback to your ears is going to be corrupted by your own voice. It is like listening closely to two messages at the same time, while also trying to produce one and synchronize them. I just don't think I can do it, LOL. To tell you the truth, I am not convinced shadowing is useful, unless you do it to pick up a kind of intonation and way of speaking. I know some say it works, but would it help recognize features like "intrusive t's" (as in prince = prints), or subtle assimilation (as in "Jom McCain" in BrE)?

The only thing I thought of that could work is "attentive listening", as I said. The more vowels and consonants you already know, the easier it will be to associate them with what you hear. New sounds will sound like they are close, but not exactly the same as the one you know, so they will sound like they are "in between". The more sounds you know, the more you are likely to interpolate correctly. This seems easy but it's not, LOL, that's why I am asking for advice here.

I guess there must be a decent answer to my question though, because I think the answer might be related to what professional impersonators or voice actors do to train. I have never known what they do.

@Estel: I sometimes hear a song on the radio that struck me because of the way that feature is emphasized, almost exaggerated. Listen to the chorus and notice the vowels in: damn, jam, AM, understand, hands.
http://www.youtube.com/watch?v=zlpNK9UKB5c

"Yes, can't repeat right away, so you need to "study" the sentences first. And when speak over the audio track you are going to "disturb" it, it's inevitable. Paying attention to an audio track and speaking at the same time is confusing... "

Johnny, this isn't true at all--take it from someone who has done some shadowing work. I suggest that perhaps you are not shadowing correctly.

You really need an audiotape, or audiobook. Listen to one sentence at a time, long enough to know its content, then speak that same sentence along with the speaker. (As mentioned before, the volume of both speakers needs to be the same so one voice won't drown out the other.)

Then, repeat this same sentence, over and over again (10-15 times isn't too many) until your pronunciation matches the target speaker's exactly. This technique works very well, but it does take time---many hours of boring, tedius, grueling work. (How long does it take to learn to play the piano?)

"tedious", rather--not "tedius". Typo...