Consider the topological space $\mathbb{R}$ with discrete metric
$$d(x,y) = \begin{cases} 1 & x\neq y\\ 0 & x=y \end{cases}$$
We know that the metric space is first countable. So for each $x\in \mathbb{R}$, there exists a countable neighborhood basis at $x$.
My question is what is that basis?
I think $\{\{x\}\}$ is a local base at each $x$, so there is only one element in this basis? How about if I consider the open ball with $r>1$, then the number of elements in this basis is infinite. It becomes uncountable.
I am confused about how to find the neighborhood basis of $x$.
You are right; $\bigl\{\{x\}\bigr\}$ is a a countable (finite, actually) system of neighborhoods of $x$, for each $x\in X$.
Note that, with respect to the discrete topology, $V$ is a neighborhood of $x$ if and only if $x\in V$.