How to show a ring is normal or not, and how to show the normalisation of the ring

I am confused about how to show whether a ring is normal or not. For example, consider the $k$-algebra $k[x,y] /\langle x^2 - y^3 \rangle$, which is a domain. How do I show it is not normal? Are there any standard techniques? I know that I want to show it is not integrally closed in its fraction field, but I can't work out how to do this, i.e. find some $z$ in the fraction field, integral over the ring but is not in the ring.

Also how do I work out what the normalisation actually is? Sorry I have no working to offer because I am completely stumped.

As wxu explained in a comment, $x/y$ is integral over your ring, which therefore cannot be normal. To compute the normalization, observe that the map $x \mapsto t^2$, $y \mapsto t^3$ sends your algebra isomorphically onto $k[t^2,t^3] = k \oplus (t^2) \subset k[t]$. This subalgebra has codimension one, whence $k[t]$ is its normalization.

If you know any algebraic geometry, this is the algebra of polynomial functions on the cuspidal cubic curve $X = \{ (x,y) \in \mathbb{A}^2 \ | \ y^2 = x^3 \}$, pictured here. A curve is nonsingular if and only if its algebra of functions is normal, and it is geometrically obvious that this curve is singular at the origin. The normalization map corresponds to the parameterization $\mathbb{A}^1 \to X$ given by $t \mapsto (t^2,t^3)$.

There is an easy way to tell where the ring $k[x,y] / f(x,y)$ is not regular: it is precisely at the ideal $$ \langle f(x,y), \partial_x f(x,y), \partial_y f(x,y) \rangle$$ This remains true in more variables and more equations -- take all first-derivatives of all equations. The geometric interpretation is that this ideal describes the singular set of your system of equations.

Since your ring is one-dimensional, being non-regular is synonymous with being non-normal.

Applying the test and assuming we aren't in characteristic 2 or 3, we get the ideal $$ \langle x^2 - y^3, 2x, 3y^2 \rangle = \langle x, y^2 \rangle$$ so the ring is not normal. The normalization will involve adjoining fractions $u/v$ of things in this ideal. Alas, I haven't gone through this procedure often enough to be able to do it systematically, so I can't offer advice on that.