The core: Riemann tensor, Bianchi identities, Einstein equation. Pay special attention to Box 14.5 (derivation from action principle).
Quadrupole formula, TT gauge, energy flux. Still a benchmark. The core: Riemann tensor
Classic material: photon sphere, event horizon, Eddington–Finkelstein coordinates. Their discussion of “river model” of black holes is unique. The core: Riemann tensor