Is there a systematic way to find bijective function between non-empty set $\mathbb{X}$ and Natural numbers? I have read about the countable and uncountable sets. By the definition of the countable set I have to find a bijection between non empty set $\mathbb{X}$ and natural numbers. I want to know is there a systematic approach to find out such functions to specify whether the desired set is countable or not? 

P.S I mean for most things I encounter, how would I know if it's countable?

Yes, finding such a function means simply ordering the elements of the set $\mathbb{X}$
(i.e. putting them in a sequence, finite or not):  $x_1, x_2, ..., x_n, ... $
If you can do that, then the set is countable.
If the sequence is finite, that depends on $\mathbb{X}$ being finite. 
