If you are happy to treat GR as the low energy approximation of a full quantum theory of gravity, then the procedure is conceptually straightforward (but can quickly become technically very difficult).
- Split the metric into a background plus perturbation, $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}.$
- Expand the Einstein-Hilbert action to quadratic order in $h_{\mu\nu}$ (or higher order if you want to look at more complicated diagrams).
- Quantize the theory for $h_{\mu\nu}$ in your favorite formulation of quantum mechanics.
This approach can be used, for example, to compute Hawking radiation into gravitons by looking at metric fluctuations around a Schwarzschild background, or the power spectrum of primordial gravitational waves by looking at quantized metric fluctuations around a quasi-de Sitter background during inflation. (For completeness, I'll add that a very important subtlety when doing calculations with quantum theory on curved spacetime -- including both Hawing radiation and inflationary perturbations -- is that there is no obvious privileged vacuum state (unlike in flat spacetime where there is a Poincaire invariant ground state), and so you need to choose an appropriate state to call the vacuum state; however this is not specific to spin-2 fields)
If we expand the action to second order in $h$, then the Lagrangian has the form $\frac{1}{2} h_{\mu\nu} \bar{\mathcal{E}}^{\mu\nu\rho\sigma} h_{\rho\sigma}$, where $\bar{\mathcal{E}}^{\mu\nu\rho\sigma}$ is a second-order differential operator depending on the background metric $\bar{g}_{\mu\nu}$. The propagator (in some gauge) is the inverse of $\bar{\mathcal{E}}$. In the geometric optics limit (when the wavelength of the perturbation is small compared to the curvature, you will find that the solutions to the classical equations of motion follow null geodesics.
The tricky part is going in the opposite direction. In other words, if you don't already know the Einstein-Hilbert action, and you want to construct a theory of a spin-2 field, where would you start? In flat spacetime, we find unitary representations of the Poincaire group which are labeled by mass and spin. Once you do that, it's possible to look for consistent interacting theories of different representations, and you find the only self-consistent interacting theory of a massless-spin 2 particles (at low energies) is GR. It's much harder to define what this procedure means, when you don't have Poincaire invariance in the background spacetime.
One approach that I like is Einstein-Cartan gravity (i.e., using the vielbein instead of the metric as the basic variable), where you can define a local Lorentz frame, and then the spin-2 field falls in the spin-2 representation of the Poincaire group in each local frame. In this approach, you can view gravity as being analogous to Yang-Mills, where the gauge group is the Poincare group, instead of $SU(N)$. (However, I am sure Wald was well aware of the Einstein-Cartan formulation at the time he wrote the passage you quoted).
