Do rational and irrational numbers flip-flop? I have found out that between every 2 rational numbers there is an irrational number, and between every 2 irrational numbers, there is a rational number.
Does this mean that the rational and irrational numbers are laid out in an alternating pattern along the real line?
 A: No. Your statements are correct, but it is incorrect to think of the real numbers like this. Because the rationals and irrationals are dense in $\mathbb{R}$, for any two distinct numbers in either set you can find another in between them (in either set, as you mentioned), so you can't possibly have a notion of 'adjacent'-ness of rationals and irrationals.
For a deeper answer you may be interested in order theory.
Here is another argument:
To have the rationals and irrationals 'alternate when lined up' in some sense, we would have to take each irrational number, and put it in between two other rational numbers. However, this would imply that for every irrational, there is a distinct rational number that is lined up to it's right, so we could make a 1-to-1 mapping from the irrationals to the rationals.
But the rationals are countable, so that would imply that the irrationals are countable too! 
But the irrationals are uncountable!
so it's impossible to line up the rationals and irrationals so that they alternate and each number only appears once.
