A good way to start "AI". Write a decision tree they will serve you well and with boosting do even better. Basic but useful stuff: logistic regression, armed bandits, weighted experts, kNearest, k means , Kernel Density estimation and Naive Bayes. That covers online, ensemble, super and unsuper vised algorithms. Goodluck!
http://slowping.com/2012/self-driving-lego-mindstorms-robot/
Yes, for some problems it might be faster and better to code logic yourself, but there are also tasks (such as pattern recognition) where NNs might be more effective.
Could anyone with expertise say if this would be enough to build a foundation? How much math background do you need?
https://www.coursera.org/course/ml (From one of the authors of this paper!)
https://www.coursera.org/course/vision
https://www.coursera.org/course/computervision
Prof. Hinton's videos are very watchable: