Skyrmions recently discovered in chiral magnets are a promising candidate for magnetic storage devices because of their topological stability, small size ($∼ 3-100$nm), and ultra-low threshold current density ($∼ 10^{6}$A/m$^2$) to drive their motion. However, the time-dependent dynamics has hitherto been largely unexplored. Here we show, by combining the numerical solution of the Landau-Lifshitz-Gilbert equation and the analysis of a generalized Thiele's equation, that inertial effects are almost completely absent in skyrmion dynamics driven by a time-dependent current. In contrast, the response to time-dependent magnetic forces and thermal fluctuations depends strongly on frequency and is described by a large effective mass and a (anti-) damping depending on the acceleration of the skyrmion. Thermal diffusion is strongly suppressed by the cyclotron motion and is proportional to the Gilbert damping coefficient $α$. This indicates that the skyrmion position is stable, and its motion responds to the time-dependent current without delay or retardation even if it is fast. These findings demonstrate the advantages of skyrmions as information carriers. ; Comment: 9 pages, 10 figures