|MadSci Network: Physics|
This situation is unstable, I'll explain why and then answer what would happen if you could have a sound emitter travelling at the speed of sound.
As objects like jet aircraft pass through the "sound barrier" they create a sonic boom. The planes used to tear themselves apart instead, because of the enormous shock wave they were flying through. If you make a sound wave, you must move air back and forth to create the pressure wave that is the sound itself. If your sound emitter is moving at the speed of sound, you will create a wave of compressed air that you are moving along with at the same speed. If your emitter is making continuous waves (a tone), then you will be kicking the pressurized air in the wave forward, compressing it together with more air in front of you. These waves build up until they tear apart your sound emitter. So if you speak while travelling at the speed of sound you will create a continuous shock wave that will tear you apart. Just moving through the air would create sound at that speed, which is sort of how planes make sonic booms as they reach the sound barrier. It really is a barrier, a pressure wavefront that you either get through or break up on. Nothing travels at exactly the speed of sound for long. The basic effect on the sound waves is that they will be doppler shifted to infinite frequency, so they will merge and cease to be distinct sound waves...they will become a continuous shock wave.
On one further note, sound spreads out spherically from its point of origin in all directions. So even if you make a sound and travel in any direction at the speed of sound, you are travelling "with the sound," as your question puts it. The direction you travel is relatively unimportant unless you hit something.
For a web page with basic explanations of sonic booms, including the cone-shape of the shock wave and a photo of a jet passing the sound barrier, visit this page on sonic booms. For a wordy explanation full of facts about the intensity of sonic booms and the damage they cause to structures and human beings, visit this NASA page.
Try the links in the MadSci Library for more information on Physics.