The wave equation in Minkowski space can be given as $-\frac{\partial^2\phi}{\partial t^2}+\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}= 0$.
In a curved spacetime this can be re-written in the form $g^{\mu \nu}\nabla_{\mu}\nabla_{\nu}\phi = 0$, where $\nabla_{\mu}$ and $\nabla_{\nu}$ are covariant derivatives and $g^{\mu\nu}$ is a metric in $\it{M}^4$.
How can I determine the form of this equation in the spacetime with line element $ds^2 = -dt^2 + a(t)(dx^2 + dy^2 + dz^2)$?
Some hints.
First of all find the metric tensor $g_{\mu \nu}$ and then its inverse $g^{\mu\nu}$, this is rather easy.
Then recall the definition of covariant derivative:
$$\nabla_{\mu}g_{\alpha\beta} \equiv \dfrac{\partial g_{\alpha\beta}}{\partial x^{\mu}} - \Gamma^{\eta}_{\mu\alpha}g_{\eta\beta} - \Gamma^{\gamma}_{\mu\beta}g_{\alpha\gamma}$$
As well as the definition of Christoffel Symbols
$$\Gamma^{\ell}_{ij} \equiv \dfrac{1}{2}g^{\ell k}\left(\dfrac{\partial g_{\ell i}}{\partial x^{j}} + \dfrac{\partial g_{\ell j}}{\partial x^{i}} - \dfrac{\partial g_{ij}}{\partial x^{\ell}}\right)$$
Then it's all about a very simple calculation, considering your metric. (P.s. don't forget about Einstein's convention over repeated indexes)