Question: How would you find the cardinality of the basis of $\mathbb{R}$ over $\mathbb{Q}$? Is it countable or uncountable? 

In general, how do you find the cardinality of a basis of an infinite-dimensional vector space? Do you just search for a bijection between the basis and, say, $\mathbb{N}$ or $\mathbb{R}$? What are some instructive examples?

Answer: 
If $V$ is a vector space over the field $F$, and $B$ is a basis for $V$ then every element in $v$ is the unique combination of finitely many elements from $B$ with coefficients from $F$.

So each $v\in V$ can be represented as a finite subset of $B\times F$ which is a function. How? $v(b)=\alpha$ if $b$ is a basis element which has a nonzero coefficient, $\alpha$, in the linear combination of $v$ from $B$.

How many such functions are there? Now let's put into play the assumption that $V$ is infinite.

At least $B\times (F\setminus\{0\})$, since each singleton $\{\langle b,\alpha\rangle\}$ corresponds to the vector $\alpha\cdot b$. But if $X$ is infinite, the set of all finite subsets of $X$ has the same cardinality as $X$. So we get a bijection using Cantor-Bernstein. And since we're talking about infinite sets, we can consider $F$ and not $F\setminus\{0\}$.

So we get that $|V|=|B\times F|=|B|\cdot|F|$. But multiplying two non-zero cardinals, at least one of them is infinite, we get that $|B|\cdot|F|=\max\{|B|,|F|\}$.

In particular, if $|V|>|F|$ then $|V|=|B|$, or in other words $\dim V=|V|$.

So now if we want to apply this to $\Bbb R$ as a vector space over $\Bbb Q$, what do we get?
The cardinality of a Hamel basis of $\mathbf R$ over $\mathbf Q$ can't be countable, since it would imply $\mathbf R$ is countable.

A more general result is that if $E$ is an infinite-dimensional Banach space over a subfield $K\subset\mathbf C$, Hamel bases have all cardinality $\lvert E\rvert$.