This two-part review summarizes observations, theory, and simulations of interstellar turbulence, highlighting its implications for various fields of astrophysics. The first part focuses on diagnostics for turbulence in the cool interstellar medium, energy sources, and turbulence theory. Observations of structure functions, autocorrelations, power spectra, and delta variance have been used to characterize turbulence. Energy sources include supernovae, superbubbles, galactic rotation, self-gravity, and cosmic ray streaming. Turbulence theory covers fluid equations, modes, inviscid quadratic invariants, scaling arguments, higher-order structure functions, energy transfer, velocity distributions, and turbulent pressure. The role of self-gravity in turbulent gas evolution is discussed, leading to new paradigms for star cluster formation, stellar mass functions, stellar rotation, binary stars, and magnetic fields. The review concludes with a reflection on progress and outstanding problems in understanding the interstellar medium.This two-part review summarizes observations, theory, and simulations of interstellar turbulence, highlighting its implications for various fields of astrophysics. The first part focuses on diagnostics for turbulence in the cool interstellar medium, energy sources, and turbulence theory. Observations of structure functions, autocorrelations, power spectra, and delta variance have been used to characterize turbulence. Energy sources include supernovae, superbubbles, galactic rotation, self-gravity, and cosmic ray streaming. Turbulence theory covers fluid equations, modes, inviscid quadratic invariants, scaling arguments, higher-order structure functions, energy transfer, velocity distributions, and turbulent pressure. The role of self-gravity in turbulent gas evolution is discussed, leading to new paradigms for star cluster formation, stellar mass functions, stellar rotation, binary stars, and magnetic fields. The review concludes with a reflection on progress and outstanding problems in understanding the interstellar medium.