Step 1
The first step in improving your reading speed when getting to the core is whenever you are reading a para, no matter the length, you should always be asking yourself - what is the point? What is the author trying to say? Is the author's message making sense? As you move through the para, keep asking yourself if the author is making the same point or has the author has switched focus. This point should lead you to the thesis statement of the writing. If the thesis is not explicit, try to figure out how you would define it yourself.
Step 2
The second step is foreshadowing - where you try to guess where the author is going with the para. Once you think you have the core - think about the next thing the author would try to do to prove their point. Now, you don't have to be right. This isn't about always knowing what somebody is going to say next. The purpose of this is to allow you to think while you're reading through the piece and think about a natural flow of information. If the author goes a different way, that's fine. Go with them, but ask yourself 'what are they trying to say?'
Step 3
A third step in training yourself to understand the core of a passage is to imagine how you would explain the passage to others who haven't read it for themselves. How would you describe the passage if you had half the space the author took in the original writing? Now, how would you describe it using 1/3 of the space? Next, go for 1/4 of the space. Your next challenge is describing the passage in four sentences. Your final challenge is to describe the passage in one sentence.
Step 4
The fourth and final step in this process is critique and evaluation. As you look at what the author seemed to be trying to say - were they successful in saying it? How could they have been more clear in their point? Were they concise enough or did they actually need more depth? How did the core match their thesis statement, if they had one?
The better you get at figuring out the core in the writing of others, the better you will become in your own writing. Getting better at figuring out the core will also improve your reading speed and reading comprehension. This is a skill that will help you in fields beyond English Composition and assist you in reading information from a variety of sources. Since the more you practice, the better you will get, challenge yourself. The best area of challenge is to read works you are unfamiliar with or are very new to you. If you can properly find the core of these pieces (confirmed by reading abstracts and publisher notes) you will know you have developed the skill to a point where it will be useful to you throughout your studies, no matter where they take you.