Recent advances in nonlinear optics and strong-field chemistry highlight the need for calculated properties of organic molecules and their molecular ions for which no experimental values exist. Both static and frequency-dependent properties are required to understand the optical response of molecules and their ions interacting with laser fields. It is particularly important to understand the dynamics of the optical response of multielectron systems in the near-IR (lambda similar to 800 nm) region, where the majority of strong-field experiments are performed. To this end we used Hartree-Fock (HF) and PBE0 density functional theory to calculate ground-state first-order polarizabilities (alpha) for two series of conjugated organic molecules and their molecular ions: (a) all-trans linear polyenes ranging in size from ethylene (C2H4) to octadecanonene (C18H20) and (b) polyacenes ranging in size from benzene (C6H6) to tetracene (C18H12). The major observed trends are: (i) the well-known nonlinear increase of a with molecular size, (ii) a significant increase of a upon ionization for larger systems, and (iii) for larger ions, the dynamic polarizability at 800 nm is much larger than the static polarizability. We have also compared the HF and PBEO polarizabilities of the linear polyenes up to octatetraene calculated with second-order Moller-Plesset perturbation theory (MP2) and coupled cluster theory with single and double excitations (CCSD). For neutral molecules the results at the PBEO and HF levels are very similar and ca. 20% higher than the MP2 and CCSD results. For molecular ions, results at the HF, PBEO, MP2, and CCSD are all very close. We discuss the size scaling and frequency dependence of a, and provide simple models that capture the origin of the change in the static and dynamic polarization upon ionization.