The Cabibbo-favored (CF) and singly Cabibbo-suppressed (SCS) two-body hadronic weak decays of antitriplet charmed baryons $\Lambda_c^+$, $\Xi_c^0$ and $\Xi_c^+$ are studied in this work. In the topologic diagram approach, both factorizable and nonfactorizable contributions are considered. The estimation of nonfactorizable contribution from W -exchange and inner W -emission diagrams relies on the pole model and current algebra. The non-perturbative parameters in both factorizable and nonfactor- izable parts are all calculated in the MIT bag model. We calculate the branching fractions and up-down decay asymmetries for all the CF and SCS decays of these antitriplet charmed baryons. The prediction of $B(\Xi_c^+ \to \Xi_0 \pi^+)$ agrees well with the recent Belle result, while the calculated $B(\Xi_c^0 \to \Xi^- \pi^+)$ is too large compared to the recent Belle measurement. We conclude that these two $\Xi_c \to \Xi \pi^+$ modes cannot be simultaneously explained within the current-algebra framework for S-wave amplitudes. This issue needs to be resolved in future study. The long-standing puzzle with the branching fraction and decay asymmetry of $\Lambda_c^+\to \Xi_0 K^+$ is resolved . We find that not only the calculated rate agrees with experiment but also the predicted decay asymmetry is consistent with the SU(3)-flavor symmetry approach in sign and magnitude.