I'm a business/international relations person and a lot of my job is flying around. I have had a lot of downtime recently, and couldn't find a sustainable hobby to fill in that time.
Until I found Michael Spivak's Calculus and decided that it was legitimately a very fun book to read. I didn't actually do the problem sets, but I read the book carefully, and can say that while I by no means mastered the material, I'm generally conversant in it. I might actually end up doing the problem sets at some point, but that's another thing…
In a similar vein to my previous endeavors, becoming "fluent" in undergraduate biology and philosophy through self-study during my downtime, I'd like to do the same thing with mathematics and statistics.
Can someone help me plan out and structure what books I should read and in what order? Let's try to avoid popular science books. I liked the level of technicality in Spivak's book. Again, I'm not trying to reach any sort of academic mastery, just technical "conversational" fluency.
There are plenty of "what should I read" questions around, but I think mine is slightly different, by virtue of asking for a structure, and specifying what I want to achieve. Also, I like the proof-based approach used by Spivak, and would like to see something similar for statistics.
edit
To clarify, when I read Spivak's book Calculus, I didn't skip the dense parts. I read and understood the proofs. Whether I could replicate them on my own is another issue–I attribute this to the lack of problem sets completed–but I enjoyed the dense parts of Spivak's books. So, I am absolutely looking for something more technical than A Brief History of Time, etc, etc.
Best Answer
Given what you've said, I recommend looking at the following.
1. The MAA's New Mathematical Library books
2. Mathematics: Its Content, Methods and Meaning
Note that #2 above, a well known English translation that first appeared in 1965 (the date of the 3-volume hardback edition I have), is now available as a relatively cheap Dover paperback. Incidentally, all 12 of the present amazon.com reviews give this book a 5-star rating (the maximum).
3. English translations of the Russian Popular Lectures in Mathematics Series. Besides the google search I embedded in the previous sentence, see also David Singmaster's list.