Which sets are by definiton countable/uncountable? Are there some general rules on whether a set is countable or not? Thus far I only know about $\mathbb{R}$ and $\mathbb{C}$ being uncountable, so I wonder if there are possibly any theorems that ascertain the countability/uncountability for sets, i.e., the cartesian product of a set, the power set of a set... etc. Is there maybe an intuitive way to see that? Other than going by the definition of countability itself.
Thank you very much!
 A: If a set $S$ contains an uncountable set, then $S$ is uncountable too. So, every set that contains, say, $\Bbb R$, is uncountable. Also, if $S$ is an infinite set, the set $\mathcal P(S)$ of all subsets of $S$ is uncountable. In particular, $\mathcal P(\Bbb N)$ is uncountable.
On the other hand, any subset of a countable set is coubtable. And uny countable union of countable sets is also countable.
A: For the Cartesian product of two sets, you basically multiply the cardinalities.  In the case of infinite cardinality, you get the larger cardinality back.  I have jumped up to infinite cardinality, since the distinction between countable and  uncountable is really just the beginning of the story.
This is just a special case of Cantor's "infinite paradise", and its cardinal arithmetic.
For instance, in the case of power sets, the cardinality of the power set of a set is always greater than the cardinality of the original set:  $|\mathcal P(A)|\gt|A|$, by Cantor's famous diagonal argument.  Particularly salient is the proof that the real numbers are not countable, since if you claim you could put them in a list, it is possible by the diagonal argument to construct a real not in the list by changing the diagonal entry on each element.
A: A set non-empty set $X$ is countable iff there is a bijection $f:\mathbb{N}\to X$ iff there is an injection $i:X\to\mathbb{N}$ iff there is a surjection $s:\mathbb{N}\to X$.
Just as another piece of information, if $A$ is a set and $B$ is a proper subset of $A$ and there is a bijection $f:A\to B$, then $A$ is infinite.
A: Basically a set $\mathcal{S}$ can be called countable, if there exists a bijective function $f$ between $\mathcal{S}$ and the set of natural numbers: $\mathbb{N}$ (i.e. $\mathcal{S} \overset{f}{\sim} \mathbb{N}$). In this case we can say that sets $\mathcal{S}$ and $\mathbb{N}$ are equinumerous, and that is any set, which is equinumerous to $\mathbb{N}$ is called countable.
