The Laplacian can be defined on any Riemannian manifold as div grad. Here grad f for f a smooth function is the vector field dual to the 1-form df via the bilinear form of the metric. Div of a vector field X corresponds to taking the covariant derivative $\nabla X$, which is a (1,1) tensor, and taking the trace of that. In local coordinates one can give a formula using the symbols for the metric, which should yield what you are looking for.
Another way to define div is to take the Lie derivative of the volume form: that is, $L_X V = (div X) V$. The volume form depends on an orientation, which can be locally chosen. This way is actually probably easier for computing in local coordinates since you don't need to worry about a covariant derivative or Christoffel symbols.
For a reference, see e.g. Taylor's Partial Differential Equations, Vol. 1. In Folland's Introduction to Partial Differential Equations, there isn't much about Riemannian manifolds, but Folland does talk about how the Laplacian changes with respect to new coordinates.