Salt and nutrient transport and transformations during water infiltration directly influence saline soil improvement and the efficient use of water and fertilizer resources. The effects of soil initial salinity (18.3 g/kg, 25.5 g/kg, 42.2 g/kg, 79.94 g/kg, and 165 g/kg, respectively, labeled S1 to S5) on the infiltration and leaching characteristics of water, salt, and nitrogen were analyzed via a one-dimensional vertical fertilizer infiltration experiment. Meanwhile, the estimation models of cumulative infiltration and wetting front, including the effect of soil initial salinity, were established. The results showed that, with the increase in soil initial salinity, the cumulative infiltration within the same time decreased, and the migration time of wet front to 45 cm was longer. The time required for S5 to reach the preset cumulative infiltration was more than six times that of S1, and, for the wet front migration to 45 cm, the time requirement for S5 was about four times that of S1. In the established Kostiakov model and wetting front model, the coefficients all decreased with the increase in soil initial salinity, and the test index R2 values both reached 0.999. In the Kostiakov model, coefficient K had a linear relationship with the natural logarithm of initial soil salt content, while index a had a direct linear relationship with initial soil salt content. The cumulative leachate volume decreased with the increase in soil initial salinity, and the corresponding data of S3 and S5 were reduced by 37% and 57.3%, respectively, compared with S1. The electrical conductivity values of S1, S3, and S5 were 15.4, 209.8, and 205.6 ms/cm, respectively, being affected by the initial content in soil, soil moisture transport rate, and exogenous potassium nitrate (KNO3) addition. The NO3−-N concentrations in the leachates of S1, S3, and S5 at the end of leaching were 55.26, 16.17, and 3.2 mg/L, respectively. Based on the results of this study, for soil with high initial salinity, the conventional irrigation amount (2250 m3/ha) of the general soil in the study area could not meet the requirements of leaching salt. These results can provide a reference for the formulation of irrigation and fertilization strategies for soils with different salinity and contribute to the sustainable development of saline soil agriculture and the ecological environment.