Ok, let's say, for my explanation's sake, that the sound can only travel along one plane (like you suggested in your question). So only towards you or away from you. Let's also assume this is happening at "room" temperature (68F) in dry air. Therefore, it is assumed that the speed of sound will be 767mph (or 1125ft/s or 343 m/s)
Say a sound is coming straight towards you. It will get to you at a rate of 767 mph. If you go straight towards the sound at the speed of sound, it will get to you twice as fast (343m/s + 343m/s = 686m/s), and you would get an amplified doppler effect.
Take a look at this equation to figure out how much the doppler will effect your sound:
(f) is the observed frequency (what you hear),
(f0) is the emitted frequency (what the original sound is),
(V) is the velocity of waves in the medium (343 in this case), and
(Vr) is the velocity of the receiver relative to the medium; positive if the receiver (you) is moving towards the source.
So, for a 1k tone: f = ((343 + 343)/343)1000
So f = 2000hz (or 2kHz). So, basically, if you travel at the speed of sound directly at a sound, everything will be exactly one octave higher.
However, if you travel at the speed of sound in the opposite direction of the sound coming towards you, the sound will never get to you, hence, you would never hear it, which is good, because I don't want to fight with any more equations...
But......If you were to go JUST SHY (say by 3m/s for this example) of the speed of sound in the opposite direction of the sound, it would look like this: (same equation)
Again, for the 1k tone: f = ((343 + (-340))/343)1000
So f = 8.746hz. Basically, your 1k tone would cease to be in your hearing range. Sound would sound like it's slowing down until it stops as you approach 343 m/s in the opposite direction of the sound.
So yes, in answer to your question - it would sound very strange and slow and all, because the sound would be arriving to you very slowly...
NOW, if you are currently going the speed of sound, and you make a sound, that sound will now be going 686 m/s relative to ground. So, anyone that hears you speak while you're going that speed will hear you at exactly 1 octave higher (unless they themselves are moving).
Even with reflections and all, this pretty much stays the same. However, temperature and medium (air, humidity, water, steel, etc...) would change the sound more drastically as the speed of sound would change.
Cool question! Let me know if you need me to clarify anything!