For a person to move at supersonic speeds, they would need to speed up everything: Their mental capacity, perception, metabolism, heat dissipation, etc.
If you don't upgrade mental capacity, then you'll splat yourself against obstacles. If you don't upgrade perception, then your eyes will see the world as a confusing mish-mash of the last few things you've looked at, as if you were tripping really hard. If you didn't upgrade metabolism, you'd burn yourself out in less than a mile. You get the idea.
Thus, to do this, you have to do something functionally identical to increasing the rate at which the person's entire body experienced time, slowing everything down, comparatively speaking.
I did a treatment of what this would be like in my latest book, "Dystopia: Clempson Goes to Hell." In that book, one character's power is to adjust her relative speed through time. In chapter five, the main character gets to experience things from an accelerated perspective. Here are the main characteristics.
Speeds
Standard jogging speed is around 5 mph. If you think you can flat-out run under this much stress, you probably haven't done much running. The speed of sound is 767 mph, so if you experience an accelerated frame rate of x150, you will be able to jog at the speed of sound.
Air gets thicker
Air is roughly 1/250th as thick as water, so at 250:1 acceleration, you are experiencing air with the thickness of water, but without the buoyancy. Search for "running under water" for what this would look like. Breaking the sound barrier wouldn't be quite this bad. If you accelerated over 250:1, you'd basically be swimming in the air, and could become airborne, but you'd have to worm your way through the air, pushing it out of the way like a snake. (this is why I named that character "Serpentine")
Gravity drops
Gravity would pull you 1/150th as fast, but would have the same force. Standing still, your legs would be supporting the same weight, but you would take longer to fall. This means your grip on the ground would be poorer, so you'd have to lean into your run to get enough grip on the ground. Again, watch the "running under water" videos.
Light would be dim, and red-shifted
At the speed of sound, you'd be getting 1/150th as much light. If you're familiar with cameras, it would be like using a shutter speed 150x faster (14 settings on most cameras). Anyone who's studied high speed photography knows how this works.
Due to red-shift, you'd be seeing far infra-red as your normal spectrum. Conversely, your body heat would be extreme UV and soft X-rays. At around 100:1, you could give a person radiation burns by standing next to them for too long.
Sound wouldn't exist
Just kidding. Hearing would be preserved far better than sight. Our vision only gives us a single doubling of wavelength, but our hearing can perceive closer to ten doublings. Thus high-pitched screeches would become low-pitched groans that last forever. At 150x, all sounds would be around 21 decibels quieter.
External impact
This is something that comic books get WAY wrong. They have silly things, like the Flash shifting between his fighting togs and normal person outfits so someone thinks there are two people standing in front of them. Nope.
You're imparting an immense amount of energy into the air. You have to move your weight in air out of the way every 200 feet or so. You'd make sonic booms anytime you reached for something. A whip crack is breaking the speed of sound with a chunk of air smaller than a cubic centimeter. Scale that up to an arm, and you're breaking eardrums. Scale it up to a whole body, and you're breaking walls.
If you're taking a casual stroll through an office, even at 50:1, you'd have a cloud of paper and dust behind you. Ignoring the sonic boom, your vacuum wake could still shatter windows, and would definitely knock people over.
