A predator–prey system with two delays and stage structure for the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. By choosing the time delay as a bifurcation parameter, the existence of Hopf bifurcation with respect to both delays is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the positive equilibrium is feasible. By using the comparison theorem, sufficient conditions are obtained for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. Numerical simulations are carried out to illustrate the main results.