if you REALLY want to be technical there is no definative answer to this question, only theories. However, some experiments have provided data that shows that if you were to travel the speed of light, then the light cone would itself travel at its normal speed and would NOT travel 2x as fast as itself. Here is an exerpt from Einsteins theory of special relativity:
In 1905 he realised how it could be that light always goes at the same speed no matter how fast you go. Events that are simultaneous in one reference frame will happen at different times in another that has a velocity relative to the first. Space and time cannot be taken as absolute. On this basis Einstein constructed the theory of special relativity, which has since been well confirmed by experiment.
Questions of relative velocity in relativity can be answered using the velocity subtraction formula v = (w - u)/(1 - wu/c^2) (see relativity FAQ: velocity addition). If you are driving at a speed u relative to me and you measure the speed of light in the same direction (w = c in my frame), the formula gives v the speed of light in your reference frame as, v = (c-u)/(1 - u/c). For any speed u less than c this gives v = c so the speed of light is the same for you. But if u = c the formula degenerates to zero divided by zero; a meaningless answer.
If you want to know what happens when you are driving at very nearly the speed of light, an answer can be given. Within your car you observe no unusual effects. You can look at yourself in your mirror which is moving with the car and you will look the same as usual. Looking out of the window is a different matter. The light from your headlights will always go at the speed of light in your reference frame. It will strike any object in its path and be reflected back. Everything else will be coming towards you at nearly the speed of light, so the light reflected from it will be Doppler shifted to very high frequencies--towards the ultraviolet or beyond. If you have a suitable camera you could take a snapshot. The objects passing are contracted in length but because of the different times of passage for the light and effects of aberration, the snapshot will show the objects you pass as rotated.
The reason, though, this can not be ever truelly proven is because we can not go the speed of light so this is a hypothetical question and thus can not have a definative answer.