Here are the main mistakes english-speakers do about sound softening in Russian:
1. Don't soften consonants followed by soft vowels or the soft sign.
2. Pronounce soft vowels after soft consonants improperly like at the beginning of a word.
There shouldn't be the Y(Й)-sound at the beginning of the vowel sounding in this case.

The only correct way of memorizing the principle of softening is to treat them as pairs.
When you come upon a soft vowel or the soft sign, remember that they've also softened the preceding consonant, like:

Мяч has the Я sound. It's pronounced similar to "Yah", only when it is standalone.
The first Y-part is only needed to put the middle of the tongue into the correct "soft" position like at the beginning of a word or something like that.

In the мяч-word the М is also softened by the Я and it's pronounced like [М'] with raised middle of the tongue. But:
This raised position is also used for the Я-sound, so:
the soft (palatalized) M prepares the tongue for the following Я-sound.
So you don't need to say that Y-part, just saying the rest when the Y is removed.
They are a pair -- the soft M and the soft Я. Same for every single soft vowel running after every single consonant,
except ones affected by those stupid spelling rules, that are used only for writing and just puzzle everyone but Russians, when they learn the language.
This way of saying when one sound flows out of the previous one is very natural, and it's used by native russians when we speak.

Getting used to this way of speaking will eliminate following mistakes with ЖШЧЩ and vowels after them, which is
another greeeat trouble for english-speakers learning Russian.

As I've noticed during the teaching process short simple and maybe childish examples work better than any explanations,
so here's the one for those who don't feel certain in the softening:
Do some self-hypnosis, imagine you're a Russian :"": cat and say МЯЯУ. That's the great simple example of the soft M and the soft Я running together,
explaining the main principle of softening in Russian: http://www.sendspace.com/file/be9yhj
Notice that the correct soft M is highly important to make the correct Я in the words.

P.S. And I should admit that our languages have only one identical sound. It's when we snore in the sleep