The velocity and attenuation of ultrasonic waves and the characteristic time for the Mountain mode relaxation in liquid and glassy methanol have been measured as a function of temperature, frequency, and pressure up to 30.4 GPa between 5 and 94 degrees C. The density, bulk modulus, structural relaxation times, and amplitudes are deduced from the acoustic data. The high-frequency limit of the shear modulus at 25 degrees C at the low end of the experimental pressure range can be derived from reported low-frequency viscosities and the acoustic relaxation times. It is found that, in addition to the Mountain mode, at least two relaxation times are required to describe the high-pressure ultrasonic data. The slowest of these times is 160 ps at 25 degrees C and 3.7 GPa; a volume of activation of 2 cm(3)/mol describes the pressure dependence of this characteristic time up to 10 GPa. The limitations set by the experimental relaxation times on dynamical simulation of the high-density fluid are briefly discussed.