> I want to get an expression for $\nabla_{a}g_{bc}$

If you are using the [Levi-Civita connection](https://en.wikipedia.org/wiki/Levi-Civita_connection), then the covariant derivative of the metric is *zero* by the *definition* of that connection.

See [this PSE question](https://physics.stackexchange.com/questions/47919/why-is-the-covariant-derivative-of-the-metric-tensor-zero).

If you don’t know which connection you’re supposed to be using, you’re almost certainly supposed to be using Levi-Cevita’s. It’s the unique torsion-free connection that preserves lengths and angles under parallel transport.