If "light" has no mass, how is it propelled?

First of all light does have mass (well sort of), the mass that is related to its velocity according to E = mc^2 light has no REST MASS, that is to say if light (photons) stop moving they stop existing. This happens when they are abosrbed.

Mass doesn't make things move, take a car for example, does it move without petrol or someone towing or pushing it? Of course not, so mass needs energy to move. Photons are pure energy.

If you want to get deeper then you'll get bogged down in what's known as the wave-particle duality of light and this is mind blowing stuff.

<< It would seem to me that there would need to be mass involved in order for something to be moved. >>

That's because you don't know what light is.

Light is oscillation of electric and magnetic fields, perpendicular to each other.

It looks like this -----> http://www.astronomynotes.com/light/emanim.gif

It's not like a ball that is accelerated by mechanical force. It's pure energy.

The light is not moving. It is waves that are propegating. Same as a wave in water, The water stays in the same place as the wave continues its travel.

light and the EM spectrum in general is a mystery it defies both major theories about it in some aspects and even fails to follow what we consider basic rules of physics under some circumstances

The same cause for the Big Bang? The force that pushes materials away? Dunno...Kinda like to know the answer myself.