  We study the prethermalization and thermalization dynamics of local
observables in weakly perturbed nonintegrable systems, with Hamiltonians of the
form $\hat{H}_0+g\hat{V}$, where $\hat{H}_0$ is nonintegrable and $g\hat{V}$ is
a perturbation. We explore the dynamics of far from equilibrium initial states
in the thermodynamic limit using a numerical linked cluster expansion (NLCE),
and in finite systems with periodic boundaries using exact diagonalization. We
argue that generic observables exhibit a two-step relaxation process, with a
fast prethermal dynamics followed by a slow thermalizing one, only if the
perturbation breaks a conserved quantity of $\hat{H}_0$ and if the value of the
conserved quantity in the initial state is $\mathcal{O}(1)$ different from the
one after thermalization. We show that the slow thermalizing dynamics is
characterized by a rate $\propto g^2$, which can be accurately determined using
a Fermi golden rule (FGR) equation. We also show that during such a slow
dynamics, observables can be described using projected diagonal and Gibbs
ensembles, and we contrast their accuracy.
