How do we prove the existence of uncountably many transcendental numbers? I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that are not algebraic).

There is a  more number-theoretic  way of looking at the matter, one that harks back to the work of Liouville.  
He showed, among other things, that a number whose decimal expansion consists of $0$s and $1$s, such that the number of $0$s between consecutive $1$s grows faster than any polynomial, must be transcendental.  
The standard example is the Liouville Constant, with the $n$-th gap being of length $n!$.
It is easy to produce continuum many transcendental numbers of the Liouville type.  
The reason the result is worth mentioning is that it arises from examination of the approximability of algebraic numbers by rationals, a subject with a long and glorious history. 
The required approximability result has a pretty quick proof that uses only high-school level mathematics.
