The easiest way to go at first (or any time you don't need a scheduled time much in advance) is to call a couple of individual golf courses and ask what they have for open tee times and do that.
All the courses' phone and addresses are listed here (this is Executive golf section) and choose the Championship menu tab for those:
Golf The Villages
Also, attend the Good Golf School class to learn about the online reservation system....they simplify the process:
http://www.golfthevillages.com/golf-...School2013.pdf