Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional.

Recall that if $V$ is a finite-dimensional vector space, then each subspace of $V$ is also finite-dimensional. So if $V$ contains an infinite-dimensional subspace, then it is infinite-dimensional. As you point out, $C[0,1]$ (the space of continuous real-valued functions on $[0,1]$) has the space $P(\mathbb{R})$ of real polynomials as a subspace. If you can show that $P(\mathbb{R})$ is not finite-dimensional, then you're done.

The plethora of answers already submitted are more than sufficient to answer the question. Interestingly, one may actually show that the collection of smooth functions on any smooth manifold forms an infinite dimensional vector space. Since $C^{\infty}(M) \subseteq C(M)$ this would also prove your result.

To prove that $C^\infty(M)$ is infinite dimensional requires a bit of work, but is actually quite intuitive. To show that a vector space $V$ is infinite dimensional, it is sufficient to show that for any $k \in \mathbb N$, one may find $k$-linearly independent vectors. Fix an arbitrary chart $\phi: U \subseteq M \to \mathbb R^n$ and take $k$-distinct points $\{x_i\}_{i=1}^k$ in $\phi(U)$. By Hausdorffness of $\mathbb R^n$, one may find open neighbourhoods $X_i$ of each $x_i$ and consequently $\epsilon_i$ such that $B_i:=\overline{B_{\epsilon_i}(x_i)} \subseteq X_i$. If $f_i:\mathbb R^n \to \mathbb R$ are bump functions on each $B_i$ and $W_i = \phi^{-1}(B_i)$, define the continuous functions $$ g_i: M \to \mathbb R, \qquad g_i(x) = \begin{cases} f_i(\phi(x)) & x \in W_i \\ 0 & x \notin W_i \end{cases}.$$ These may be extended to smooth functions $\hat g_i$ on $M$ such that $g_i|_{W_i} = \hat g_i|_{W_i}$, and these $\hat g_i$ (of which there are $k$) are easily seen to be a linearly independent set in $C^\infty(M)$.

The fact that there are infinite amounts of continuous functions is not enough. In $\mathbb{R}^2$ there are infinite amounts of vectors but it's still finite dimensional. One way to prove this is to exhibit an infinite set of continuous functions that are linearly independent. As a suggestion, consider polynomials.