
If you just throw everything into a neural network, then you won't really understand the breadth of the problems you're solving, and you'll be therefore ignorant of the limitations of your hammer. While NNs are incredibly useful, I think a deep understanding of the core problems is essential to know how to use NNs effectively in a particular domain.
After getting a grip on those concepts, Szeliski's Computer Vision: Algorithms and Applications (http://szeliski.org/Book/) had some really amazing coverage of CV in practice. Mastering OpenCV (https://www.amazon.com/Mastering-OpenCV-Daniel-Lelis-Baggio/...) was very useful when actually implementing some algorithms.
I hope someone finds this useful.
For those who want a summary,
Proof Technique
(a)Velleman's "How to Prove It" (b)Gries and Schneider's "A Logical Approach to Discrete Math"
Math
(c) Calculus (best "lite" book - Calculus by Strang (free download), best "heavy" books - (d) Calculus by Spivak, (e) Principles of Mathematical Analysis a.k.a "Baby Rudin")
(f) Discrete Math (ALADM above + (g) a good book on Algorithms, Cormen will do - though working through it comprehensively is ... hard!
(h) Linear Algebra (First work through Strang's book, then (i) Axler's)
(j) Probability (see Bradford's very comprehensive recommendations) and
(k) Statistics (I would reccomend Devore and Peck for the total beginner but it is a damn expensive book. So hit a library or get a bootlegged copy to see if it suits you before buying a copy, see brad's list for advanced stuff.)
(l) Information Theory (MacKay's book is freely available online)
Basic AI
(m)AIMA 3d Edition (I prefer this to Mitchell)
Machine Learning
(n) "Pattern Recognition and Machine Learning" by Christopher Bishop,
(o)"Elements of Statistical Learning" (free download).
(p) Neural Network Design by Hagan Demuth and Kneale,
(q) Neural Networks, A Comprehensive Foundation (2nd edition) - By Haykin (there is a newer edition out but I don't know anything about that, this is the one I used)
(r) Neural Networks for Pattern Recognition ( Bishop).
At this point you are in good shape to read any papers in NN. My reccomendations - anything by Yann LeCun and Geoffrey Hinton. Both do amazing research.
Reinforcement Learning
(s) Reinforcement Learning - An Introduction by Barto and Sutton (follow up with "Recent Advances In reinforcement Learning" (PDF) which is an old paper but a GREAT introduction to Hierarchical Reinforcement learning)
(t) Neuro Dynamic Programming by Bertsekas
Computer Vision
(u) Introductory Techniques for 3-D Computer Vision, by Emanuele Trucco and Alessandro Verri.
(v) An Invitation to 3-D Vision by Y. Ma, S. Soatto, J. Kosecka, S.S. Sastry. (warning TOUGH!!)
Robotics.
(w) Probabilistic Graphical Models: Principles and Techniques (Adaptive Computation and Machine Learning) - not about robotics per se but useful to understand the next book
(x) Probabilistic Robotics (Intelligent Robotics and Autonomous Agents) by Thrun, Burgard and Fox
PS: I own all these books (except AIMA 3 for which I only have pre pub pdfs) and if any HN folks in Bangalore want to browse before they buy anything (friggin expensive when you add amazon's postage) send me email.
PPS: on languages, I think Bradford is on the money with regards to reccomending functional languages. I would just say, also know C well. Saved my ass a few times.