Millisecond pulsars (MSPs) have been established as a class of high-energy ($\geq$0.1 GeV) emitters via the detection of significant pulsed signals, at the radio periods, of many MSPs with the Large Area Telescope (LAT) aboard the Fermi Gamma-ray Space Telescope. The detection of high-energy emission from eight globular clusters (known or suspected of containing many MSPs) that display pulsar-like spectra and detection of several new radio MSPs coincident with unassociated LAT sources further suggests that gamma-ray production must be the rule in MSPs, not the exception. Most MSP gamma-ray light curves display sharp peaks indicative of thin accelerating gaps, suggesting copious pair-creation in the open volume. MSP gamma-ray and radio light curves have been simulated using geometric outer-gap (OG), slot-gap/two-pole caustic (TPC), and pair-starved polar cap gamma-ray models and either a hollow-cone beam or altitude-limited, outer-magnetospheric gap radio model, all assuming a vacuum retarded dipolar magnetic field geometry. A Markov chain Monte Carlo maximum likelihood technique has been developed to find the best-fit model parameters for nineteen MSPs using data from the LAT and various radio observatories. Confidence contours have been created in pulsar viewing geometry which are compared with constraints from radio and X-ray observations. Model-derived beaming factors allow for more accurate determinations of gamma-ray luminosities. The best-fit viewing angles follow a uniform, angular distribution. The distribution of magnetic inclination angles favors all angles equally, contrary to analyses of non-recycled pulsars, which supports the theory that MSPs have been spun-up via accretion. There are suggestions that the radio emission should occur nearer the light cylinder. These results have implications for MSP population simulations and for addressing MSP contributions to diffuse backgrounds. The likelihood significantly favors one model over another for seven MSPs, with the TPC model largely preferred. An implied transition in the gamma-ray luminosity versus spin-down power trend is observed but more statistics are necessary to describe it.