We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds Fn = SU(n + 2)/S(U(n) × U(1) × U(1)). For all n > 1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of ℂPn + 1. The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic Kähler manifold. There is also an invariant nearly Kähler structure, because Fn is a 3-symmetric space. We explain the relations between the different structures and their Chern classes, and we prove that Fn is not geometrically formal.