I have found the upper bound for the set of all norms, but I'm unsure as how to find the lower bound. I have already shown that the cardinality set of continuous functions from $\mathbb{R}^n \to \mathbb{R}$ is the continuum
Suppose $\|\|$ is a norm on $\mathbb{R}^n$ and defined the function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ by $f(x)=\|x\|$. We prove that $f$ is continuous. Let $\{x_n\}$ be a sequence in $\mathbb{R}^n$ such that $\lim_{n \to \infty} x_n=x$, i.e. $\|x_n-x\| \to 0$ as $n\to \infty$.
To prove that $f$ is continuous it is enough to show that
$$\lim_{n \to \infty} f(x_n)=f(x)$$
Using the revered triangle inequality we have
$$\|f(x_n)-f(x)\|=\| x_n\| -\|x\| \leq \| x_n -x \| \rightarrow 0$$
Hence
$$\|f(x_n)-f(x)\| \to 0$$
which means $$\lim_{n \to \infty}f(x_n)=f(x)$$
So the cardinality of the set of all norms is at most that of the set of continuous functions from $\mathbb{R}^n \to \mathbb{R}$.
