The random coloured graph $G_c(n,p)$ is obtained from the Erdős-Rényi binomial random graph $G(n,p)$ by assigning to each edge a colour from a set of $c$ colours independently and uniformly at random. It is not hard to see that, when $c = Θ(n)$, the order of the largest rainbow tree in this model undergoes a phase transition at the critical point $p=\frac{1}{n}$. In this paper we determine the asymptotic order of the largest rainbow tree in the \emph{weakly sub- and supercritical regimes}, when $p = \frac{1+\varepsilon}{n}$ for some $\varepsilon=\varepsilon(n)$ which satisfies $\varepsilon = o(1)$ and $|\varepsilon|^3 n\to\infty$. In particular, we show that in both of these regimes with high probability the largest component of $G_c(n,p)$ contains an almost spanning rainbow tree. We also consider the order of the largest rainbow tree in the \emph{sparse regime}, when $p = \frac{d}{n}$ for some constant $d >1$. Here we show that the largest rainbow tree has linear order, and, moreover, for $d$ and $c$ sufficiently large, with high probability $G_c(n,p)$ even contains an almost spanning rainbow cycle.