Do you have a concrete example of (1)? For example, it is a theorem (called Fermat's Little Theorem) that if a is an integer and p is a prime, then a^(p)-a is divisible by p. For example, if a=2 and p=3 then 2^(3)-2 = 6 which is indeed divisible by 3. Fermat's Little Theorem states that this always works.

So, how would you take the statement and break it into several parts?

For (2), what do you mean "a reasonable amount of time". Pretty much all of mathematics is just the study of the implications of Zermelo Fraenkel set theory. So in that sense, it is far from trivial to identify all the implications of a statement and it can be a never-ending process. But perhaps I'm misunderstanding what you mean here.

And (3) sounds like a paraphrasing of "I cannot easily figure out to prove a desired statement from a set of assumptions". Again, this is describing all of mathematics. There are "desired end's" which took centuries and the combined efforts of hundreds of mathematicians to reach. And every time we prove something new it seems that we end up with more unsolved questions than when we started.

That said, the skills you're talking about (if I'm understanding you at all), are the skills that Mathematicians employ daily and continue to develop throughout their lifetime. If you're interested in developing your ability to engage in mathematical thinking, then Velleman's "How to Prove It" might be a good place to start.