Over the past few decades, numerous geochemical and geological observations of early Mars have suggested the presence of surface river systems during the late Noachian and early Hesperian. A large volume of liquid water combined with stable and long-term fluvial activity is required to explain these river systems, suggesting that early Mars must have experienced a long period during which the climate was sufficiently warm and wet for stable surface liquid water and precipitation to occur. We performed 3-dimensional climate simulations for the late Noachian and early Hesperian using a paleo-Mars global climate model (PMGCM), assuming a CO2/H2O/H2 atmosphere under the “Faint Young Sun” condition and using the assumed topography before the formation of the Tharsis load. Cases of surface pressures of up to 2 bar and obliquities of 20°, 40°, and 60° were investigated. We also developed a river transport model, CRIS (Catchment-based RIver Simulator), which enabled us to estimate global river discharge and the transport of bedload and suspended load sediments from the PMGCM output for a variety of sediment particle sizes. Our simulation results revealed that CO2 atmospheres with surface pressures of above 2 bar and a few percent of H2 composition produced a hospitable surface environment that could support the stable liquid water on the surface for long periods. Annual precipitation increased with surface pressure, and intense precipitation occurred around the low to mid-latitudes, where the majority of the valley networks are currently observed. Our river model CRIS produced river runoff, which agrees with some parts of the observed valley distribution, and revealed that these valleys could be formed in ~104 years, which agrees with previous geological studies. However, our simulation results also suggested that Martian valleys were not all created as a result of precipitation-fed river activity, and other mechanisms such as seasonal or transient snow melting should be simultaneously considered to explain Martian valley networks.