A simple example of an uncountable set that is not $\mathbb{R}$
The set of all subsets of $\mathbb N$, also called the powerset of $\mathbb N$. It is denoted by $P(\mathbb N)$ and has a cardinality of $\beth_1=2^{\aleph_0}$ which is uncountably infinite.
The set of functions from $\mathbb{N}$ to $\mathbb{N}$, often denoted $\mathbb{N}^{\mathbb{N}}$, is uncountable. This follows from using an adapted Cantor's diagonalisation argument (or by using cardinal arithmetic, but you want to avoid that).