What should you use 1. in the final Mod and 2. for debugging, 

LogManager.getLogger().log(text);

or

System.out.println(text);

and why?

for simple testing if a method is called is System.out enough,
the Logger first gives information about the time and the thread, which is sometimes useful for debugging
and second if you want to check some parameters the Logger is the better option than System.out, since you can do things like that:

public void foo(int x, int, y, int z) {
	LogManager.getLogger().log("Do foo at X:{}, Y:{}, Z:{}", x, y, z);
	//some code
}

i also would recommend you to use the Logger instead of System.out, simply to match with vanilla