A caution here...

Shannon's information theory is indeed a great intellectual achievement, and is enormously useful in the fields of data compression and error correcting codes, but there is a tendency for people to try to apply it to beyond the areas where it is useful.

Some of these applications are ok but not essential.  For instance, some people like to view maximum likelihood estimation of parameters as minimizing relative entropy. If that helps you visualize it, that's fine. But it doesn't really add anything to just directly visualizing maximization of the likelihood function. Mutual information can sometimes be a helpful thing to think about.  But the deep theorems of Shannon on things like channel capacity don't really play a role here.

Other applications seem to me to be self deception, in which the glamour of Shannon's achievement conceals that there's really no justification for some supposed application of it.  

Some of Jaynes' work is in this category. One example is his (early? he may have later abandoned it...) view that "ignorance" should be expressed by a probability distribution that maximizes entropy, subject to constraints on the observed expectations of certain functions.  This is "not even wrong". Jaynes viewed the probability distributions as being subjective (ie, possibly differing between people).  But he viewed the observed expectations as being objective.  This is incoherent.  It's also almost never relevant in practice.

The idea seems to have come about by thinking of statistical physics, in which although in theory measurements of quantities such as temperature are random, in practice, the number of molecules involved is so enormous that the temperature is in effect a well-defined number, representing a expectation with respect to the distribution of states of the system.  It is assumed that this is somehow generalizable to thought experiments such as "suppose that you know that the expected value from rolling a loaded die is 3.27, what should you use as the distribution over possible dice rolls...".  But how could I possibly know that the expected value is 3.27 when I don't know the distribution?  And if I did (eg, I recorded results of many rolls, giving me a good idea of the distribution, but then lost all my records except the average), why would I use the maximum entropy distribution?  There's just no actual justification for this.  The Bayesian procedure would be to define your prior distribution over distributions, then condition on the expected value being 3.27, and find the average distribution over the posterior distribution of distributions.  There's no reason to think the result of this is the maximum entropy distribution.

I highly recommend An Introduction to Information Theory: Symbols, Signals and Noise by John R. Pierce. Several things that are great about this book:

• it's short and concise, like many Dover books
• focuses on helping you build models with gears about how information and related concepts works
• has a bit of a bias towards talking about signals, like over a phone line, but this is actually really helpful because it's the original application of most of the information theory metaphors
• dives into thermodynamics without getting bogged down in a lot of calculations you'll probably never do

I think you'll also appreciate that it is self-aware that information theory is a big deal and a deep concept that explains a lot of stuff, and has some chapters on some of the far reaching stuff. There's chapters at the end on cybernetics, psychology, and art.

This was one of the books that had the most impact on me and how I think and I basically can't recommend it highly enough.

Try David MacKay's textbook.

I really love the essay Visual Information Theory

The book Silicon Dreams: Information, Man, and Machine by Robert Lucky is where I got mine. It's a pop science book that explores the theoretical limits of human computer interaction using information theory. It's written to do exactly the thing you're asking for: Convey deep intuitions about information theory using a variety of practical examples without getting bogged down in math equations or rote exercises.

Covers topics like:

• What are the bottlenecks to human information processing?
• What is Shannon's theory of information and how does it work?
• What input methods exist for computers and what is their bandwidth/theoretical limit?
• What's the best keyboard layout?
• How do (contemporary, the book was written in 1989) compression methods work?
• How fast can a person read, and what are the limits of methods that purport to make it faster?
• If an n-gram Markov chain becomes increasingly English like as it's scaled, does that imply a sufficiently advanced Markov chain is indistinguishable from human intelligence?

A lot of his question is to what extent AI methods can bridge the fundamental gaps between human and electronic computer information processing. As a result he spends a lot of time breaking down the way that various GOFAI methods work in the context of information theory. Given the things you want to understand it for, this seems like it would be very useful to you.

Shannon's original paper is surprisingly readable. Huffman coding is a concrete algorithm making use of information theory, which itself demonstrates information-theoretic principles, as are error correcting codes.

IMO the main concept to deeply understand when studying information theory is the notion of information content/self-information/Shannon-information. Most other things seems to be applications or expansions on this concept. For example entropy is just the expected information content when sampling from a distribution. Mutual information is the shared information content in two distributions. KL-divergence describes how much information content you're getting relative to your choice of encoding. Information gain is the difference in information content after and before you drew a sample.

For this I would recommend this essay written by me. I would also recommend Terrence Tao's post on internet anonymity. Or if you've seen Death Note, Gwern's post on the mistakes of Light. Also this video on KL divergence. And this video by intelligent systems lab.

Arbital actually has a bunch of pages on this kind of thing!

The Information by James Gleick is an interesting book on information theory.

I would recommend The Origin of Wealth, by Eric D. Beinhocker. To relate very specifically to your path here, Chapter Fourteen is an update of Micholas Georgescu-Roegen’s 1971 paper “The Entropy Law and the Economic Process”. The entire book is a fantastic telling of how the economy can be modeled as an evolutionary system.

I enjoyed this series of lectures available on youtube.

I like the professor, and tend to form better intuitions from the lecture format than from reading a textbook. In total, it would take about 15 hours to go through the course, possibly less with a bit of skipping/fastforwarding. I don't think this lecture series would give you a deep understanding of information theory, but it worked well to build my intuitions and so is a good starting point.

To start, I propose a different frame to help you. Ask yourself not "How do I get intuition about information theory?" instead ask "How is information theory informing my intuitions?" 

It looks to me like it's more central than is Bayes' Theorem, and that it provides essential context for why and how that theorem is relevant for rationality.

You've already noticed that this is "deep" and "widely applicable." Another way of saying these things is "abstract," and abstraction reflects generalizations over some domain of experience. These generalizations are the exact sort of things which form heuristics and intuitions to apply to more specific cases.

To the meat of question: 

Step 1) grok the core technology (which you seem have already started)

Step 2) ask yourself the aforementioned question.

Step 3) try to apply it to as many domains as possible

Step 4) as you come into trouble with 3), restart from 1).

When you find yourself looking for more of 1) from where you are now, I recommend at least checking out Shannon's original paper on information. I find his writing style to be much more approachable than average for what is a highly technical paper. Be careful when reading though, because his writing is very dense with each sentence carrying a lot of information ;)

Consider learning game theory.