Star Treck's warp drive assumes being able to establish a warp bubble around the space vessel to move slower than light in the local space, but faster than light in the surrounding space, to work around the limitations of special relativity.
A warp bubble would show some similarity to a worm hole. To get an idea of spacetime distortion caused by a wormhole you may like to watch some of the videos the University of Tübingen (Germany) is hosting.
The same site provides a simulation of a walk through the city with just below the speed of light.
In both cases you get visual geometric distortions.
But you get changes of the color by the relativistic Doppler shift, too, as simulated here.
When travelling faster than light you would get additional strange effects, depending on the method you overcome the speed of light barrier.
One well-known effect is Cherenkov radiation caused by charged particles travelling faster than the speed of the light in the medium, which may still be slower the vacuum speed of light. Vacuum speed of light in a flat spacetime can only be achieved by massless particles like photons.
Particles moving faster than the vaccum speed of light (tachyons) have been hypothesized, but have not yet been found.
Overcoming the speed of light would resemble somewhat falling into a black hole. When faster than light, light cannot reach you from the back. So this region would look black.
Travelling faster than light in a medium is physically feasible, at least for subatomic particles. Travelling with the vacuum speed of light or faster leads to serious problems, with the exact vacuum speed of light being the biggest challenge, since you either need to get rid of your rest mass, if travelling in flat space, or you need to warp spacetime. The latter needs huge amounts of energy causing devastating damage to the environment.
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