I think there is a big difference between working as a programmer and working in Machine Learning. It is easy for a programmer to approach machine learning as a programming problem, but it is quite different.
Programmers tend to develop code to do very specific things. An example is creating a web site with certain functionality. If the programmer is lucky, they will be given very clear requirements and produce a system that can be tested and will fulfill all the requirements. The requirements and the accuracy of the solution are often black and white. Either it works or it doesn't.
Machine Learning problems are quite different. There are few requirements. There is a goal. It might be to increase sales using a recommendation system (like Amazon). It turns out that this there is no single clearly defined solution or way of knowing that a very good solution has been found. There are only approaches to creating recommendation systems. The measurement of success might be the increase in sales, but it is difficult to know if you have found a near-optimal solution.
So, from a high level, both the problem and solution are very different for these two roles. More concretely, the tools are very different as well. A programmer can use low level tools, like C++ or even Assembly Language. I don't think people working on Machine Learning problems use low level languages. They use high level tools, like Matlab or R. They use libraries that have been developed to facilitate many different approaches to solving a Machine Language problem.
Statistics is the other side of the Machine Learning coin. Although very useful, I don't think it is really necessary to learn a lot of statistics to work in the field of Machine Learning.
So, to (finally) answer your question, it would be more useful to learn some programming and work on actual Machine Learning problems. I found that http://www.kaggle.com/ is an good source of problems to work on.
Here is a thread on Statistics Stack Exchange about accessible books for Machine Learning: http://stats.stackexchange.com/questions/18973/can-you-recommend-a-book-to-read-before-elements-of-statistical-learning
By the way, to get more information about Machine Learning problems check out Machine Learning Stack Exchange at http://machinelearning.stackexchange.com/ .
Also, if you want to learn more about Machine Learning I highly recommend the Stanford online class which is starting soon: http://jan2012.ml-class.org/ .