Bradford Cross has posted an awesome blog post (edit: removed link, since Bradford took down the post) titled "Learning about Statistical Learning". If you plan to work in ML, read the post, buy some of the books and work through them.
Could save you years of work if you are systematic from the beginning (I wasn't), especially if you are self taught (I am).
I work on different domains (Robotics/Computer Vision/Simulation) from Bradford and so have a different list of books. Please read Bradford's lists first. This is a supplement to his awesome post rather than a replacement.
I assume you are a good developer and you have a solid grip on algorithm analysis etc.(though, that said, see reccomendations for Discrete Math books below)
The first step..
Learn proof techniques *first*. You'll make no serious progress till you do. The best book is
Velleman's "How to Prove It" - reccomended by Bradford but I am repeating it here because this is critical.
In my experience you need to be somewhat comfortable with 6 branches of Mathematics before you can tackle ML. Imo, best to take a year and get these right before venturing into ML proper. (I know, it sounds awfully boring. I wasted a lot of time trying to shorten this step. In this case, the long way is the real shortcut)
(1) Calculus - best "lite" book - Calculus by Strang (free download) ,
best "heavy" books - Calculus by Spivak, Principles of Mathematical Analysis a.k.a "Baby Rudin"
(2) Some book on Discrete Math (don't know what to recommend here - I don't like Rosen's book) + a good book on say Introduction to Algorithms by Cormen et al will do [*]
(3) Linear Algebra (First work through Strang's book, then Axler's)
(4) Probability (Bertsekas is a good book for those with no prior exp) and
(5) Statistics (I would recommend 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.)
(6) Information Theory (MacKay's book is freely available online)
Brad suggests Mitchell's book.
I think AIMA (3d Edition) is much better. ( I am biased. I wrote and maintained the Java code for a long while -- children, don't do this. Java is an terrible language to develop AI algorithms in. If you need the JVM use Scala or Clojure -- and I think it covers a lot more than Mitchell does. Take a look at both. Pick one).
NB: you need all the linear algebra, calculus etc worked through before you hit this point
"Pattern Recognition and Machine Learning" by Christopher Bishop,
*then* "Elements of Statistical Learning" (free download).
Neural Network Design Hagan Demuth and Beale,
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)
and 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 (again this is just stuff *I* happened to specialize in for various projects, so feel free to ignore)
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
Neuro Dynamic Programming by Bertsekas
Introductory Techniques for 3-D Computer Vision, by Emanuele Trucco and Alessandro Verri.
An Invitation to 3-D Vision by Y. Ma, S. Soatto, J. Kosecka, S.S. Sastry. (warning TOUGH!!)
I know only about the software/algorithms side of Robotics and that too only Probabilistic Robotics. I don't know anything about hardware, electronics or Physics.
Probabilistic Graphical Models: Principles and Techniques (Adaptive Computation and Machine Learning) (strictly speaking not a robotics book, but a lot of the theory in this book is behind the algorithms in the next book
Probabilistic Robotics (Intelligent Robotics and Autonomous Agents) by Thrun, Burgard and Fox (trivia Thrun also wrote the Robotics chapter in AIMA - did I tell you AIMA rocks as a first introduction to AI?)
And that's all folks. Happy hacking!
[*] working though Cormen et al is a humungous task and can easily consume a year or more of work. Something like Sally Goldman's new Algorithm book maybe more suited to programmers.
PS: I have been getting a lot of email asking *how* one should learn X or Y. I have no idea really. The above is a list of books that worked for me and is provided only in the spirit of "these are good books that worked for me I don't know if they'll work for you."
As to how I learned, I just read books and papers, try to understand, (a lot of banging head against wall at this point) and try to solve problems and code stuff. Beyond that I have no advice on how to learn effectively etc. I am entirely self taught and have no idea how to teach this stuff. You probably need to talk to a good prof.