Speed and agility have been described as crucial abilities for soccer players. The purpose of this study was to analyze, in detail, the variance in speed and agility tasks explained by lower-body power after controlling for age and body composition. The participants were 96 male soccer players aged 16.1 ± 1.6 years. Body composition (stature, body fat percentage—BF%, body mass, and fat-free mass—FFM), lower-body power (countermovement jump—CMJ, and squat jump—SJ), speed (5-, 10-, and 35 m sprints), and agility (t-test) were assessed. Among body composition parameters, BF% presented the highest number of significant relationships with speed and agility, with the strength of correlations ranging from small (5 m sprint, r = 0.25) to large (35 m sprint, r = 0.52). The strongest correlation coefficient emerged between FFM and the 35 m sprint (r = −0.65). Significant correlations were found between vertical jump performance and the 35 m sprint (CMJ: r = −0.68; SJ: r = −0.69), followed by the t-test (CMJ: r = −0.35; SJ: r = −0.47). The hierarchical multiple regression model could explain 22% to 67% of the variance observed in agility scores and speed. BF% remained the most statistically significant negative predictor of all regression models. The CMJ remained a statistically significant positive predictor of the 35 m sprint (β = −0.581, p ≤ 0.01) after controlling for age and body composition. Integrating programs targeting lower-body power might be important to enhance speed and agility performance in youth soccer. On the other hand, future research based on multidisciplinary approaches to investigate the effects of nutritional strategies in reducing or preventing gains in BF% is still needed, which remained a significant predictor of sprint and agility performance in the final models.