The simplest fact that comes out of relativity is this:-
The speed of light is a constant, for ALL observers, no matter how they are travelling.
Therefore, it doesn't matter how fast you travel, or what speed or position you are relative to anyone else, you will always measure light travelling at the same speed.
The thing that is non-intuitive is to understand HOW light speed is a constant. The reason light speed always appears the same to every observer is because their relative chunks of space and time are NOT constant.
If you are travelling towards another observer then you will both measure light speed as normal.
If you were both travelling towards each other at the speed of light then your combined velocity CANNOT be 2c, so you still only measure light as a constant speed.
For an observer at any other position, they will see the light of you passing as normal.
Speed is just distance/time or space/time. So, if light speed remains constant then it means the ratio of space to time for each observer must change. Any body in motion will undergo a contraction of space and dilation of time. The faster you go the slower your time goes, relative to another observer, and the smaller your space. Clocks get slower and rulers shrink.
The even more crazy thing is that only observers moving at the same speed and in the same direction, can ever agree on those values. Two different observers will measure different space/time ratios for each other as it is all RELATIVE, hence the Theory of Relativity.
Einstein postulated this by thinking of basically what the OP wrote; what would it be like to ride on a beam of light? The answer is that your time would slow down to infinity and space would contract to a point, so a journey between any two points in space would appear to take zero time, to the observer on the beam of light.
This has been proven countless times by experiment.
Here endeth Physics class 101.
EDIT: Ah yes, on the last point. It's correct that an object with any mass CANNOT reach light speed. In order to increase speed (accelerate), you have to add energy. E=MC2 states that Mass and Energy are equivalent; energy has mass. Therefore the more energy you put in, the more mass an object has. As you approach light speed it takes more and more energy to accelerate a given amount. To get an object with mass to the speed of light would hence take infinite energy to do so. Only massless objects can travel at the speed of light.
