We investigate the interplay between spin and orbital correlations in monolayer and bilayer manganites using an effective spin-orbital t-J model which treats explicitly the e_g orbital degrees of freedom coupled to classical t_{2g} spins. Using finite clusters with periodic boundary conditions, the orbital many-body problem is solved by exact diagonalization, either by optimizing spin configuration at zero temperature, or by using classical Monte-Carlo for the spin subsystem at finite temperature. In undoped two-dimensional clusters, a complementary behavior of orbital and spin correlations is found - the ferromagnetic spin order coexists with alternating orbital order, while the antiferromagnetic spin order, triggered by t_{2g} spin superexchange, coexists with ferro-orbital order. With finite crystal field term, we introduce a realistic model for La_{1-x}Sr_{1+x}MnO_4, describing a gradual change from predominantly out-of-plane 3z^2-r^2 to in-plane x^2-y^2 orbital occupation under increasing doping. The present electronic model is sufficient to explain the stability of the CE phase in monolayer manganites at doping x=0.5, and also yields the C-type antiferromagnetic phase found in Nd_{1-x}Sr_{1+x}MnO_4 at high doping. Also in bilayer manganites magnetic phases and the accompanying orbital order change with increasing doping. Here the model predicts C-AF and G-AF phases at high doping x>0.75, as found experimentally in La_{2-2x}Sr_{1+2x}Mn_2O_7.
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