I find that I cannot easily manipulate statements and fit them together to potentially form something novel. For example, I cannot readily see how one would break down a theorem into parts that would together result in the original theorem. It would take me hours to see the non-trivial implications of a sufficiently complex block of logic, too. It takes even longer for me to understand how parts of different statements fit together to lead to a statement. In other words, I have three deficiencies I would like to see improved:

1. I cannot identify how to reduce a statement into several parts in multiple ways in a reasonable amount of time.
2. I cannot identify the implications of a statement in a reasonable amount of time.
3. I cannot easily figure out how to sequence statements to get a desired end.

However, I am not sure how one would go about practicing the three skills above so that they would carry over to unfamiliar areas of mathematics and benefit me there if that isn't a given. So my question is, how does one practice the three skills above effectively and generally?
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.