I know that it is technically impossible to reach the speed of light since the mass of the object traveling would reach infinity. Despite this if it were possible, would there be some sort of theoretical equivalent to a sonic boom that would occur? Since sonic booms are formed when sound waves can't catch up with an object, could the same happen with light waves? Would it look like an immense burst of light in the sky?

Technically, you can travel faster than light as long as you do it in a medium other than a vacuum. Light travels slower in these mediums, and the speed limit only pertains the the speed of light in a vacuum. When this occurs with charged particles, you get Cherenkov radiation, Which presents itself as a kind of blue glow, and yes its is somewhat of a "photonic boom".

An important thing to consider is that sound travels at a constant speed relative to the medium regardless of how fast the object creating the sound is going.

 

Light, on the other hand travels at a constant speed relative to it's source and requires no medium.

 

So an object can go faster than sound and nothing will hear that object until the object passes. On the other hand an object traveling at any speed will emit light traveling away from it.

 

Light, on the other hand travels at a constant speed relative to it's source and requires no medium.

 

 

That should be "independent of its". Light travels at a constant speed relative to all inertial frame observers, not just the source. (your statement might be interpreted too narrowly)