Is it true that for any uncountable set $\displaystyle S$, $\displaystyle S \times S$ is uncountable, and therefore this leads to the claim that $\displaystyle \mathbb{R \times R}$ is uncountable?
Yes, this is correct. If we were to assume $\displaystyle \mathbb{R} \times \mathbb{R}$ is countable, it would be easy to conclude that $\displaystyle \mathbb{R}$ is also countable (in contradiction, of course), since:
$\displaystyle f:\mathbb{R \to R \times R}$ defined by $\displaystyle f(x) = (x,0)$ is an injection, and therefore $\displaystyle \mathbb{|R| \leq |R \times R|}$
I would not be able to post anything on this forum without Wikipedia...
Namely, the section about cardinal arithmetic defines cardinal multiplication as the cardinality of the Cartesian product of the given sets. It also says that if the axiom of choice is accepted (which it usually is in regular mathematics), then for two non-zero cardinals $\displaystyle \kappa$ and $\displaystyle \mu$, if at least one of them is infinite, then $\displaystyle \kappa\cdot\mu=\max(\kappa, \mu)$, which, interestingly, equals $\displaystyle \kappa+\mu$.