Yep your intuition is good -- a lot of the mathematical techniques used in Machine Learning are motivated/derived/understood from Statistics. I'd recommend starting with a broader approach to this topic. In particular, there are often simpler heuristic approaches to a problem that are reasonably good and worth trying before trying to build a full-blown ML system. What I've learned is that knowing where ML systems fail/are overkill is just as important as knowing when to use them/build them.
If you have already brushed up on linear algebra and calculus (which you're gonna need if you want to do any serious ML) take a look at Hastie et al's Elements of Statistical Learning.
The PDF is free, and the book is both extremely well written and super comprehensive.
http://www-stat.stanford.edu/~tibs/ElemStatLearn/
You might also want to check out R, as its an amazing statistics language which has hundreds of packages available for ML. There's a large user community, and the really obscure error messages you get will teach you a lot about statistics. http://cran.r-project.org/
Also, a lot of machine learning is getting the data into a usable form, so learn how to use Unix command line tools such as sed, awk, grep et al. They are absolute lifesavers.
