It seems to me that means gravitational waves move at the speed of light. And it seems this implies that a similar theory could be drawn up for gravity as Maxwell did for the electric and magnetic forces.

This is kind of on the right track, but a bit misguided. It's not surprising that both gravitational waves and electromagnetic waves move at the same speed, because their corresponding theories (general relativity, electromagnetism) are built on top of special relativity, which tells us spacetime comes with a fundamental speed.

This does suggest that we might be able to combine the two without much trouble, and indeed that's been done long ago. Classical gravity with electromagnetism is simply described by the action $$S = \int d^4x \sqrt{-g} \, \left( \frac{R}{16 \pi} - \frac14 F^{\mu\nu} F_{\mu\nu} \right).$$ More exotically one could consider a Kaluza-Klein theory, where you don't even put in the electromagnetic field separately, but get it automatically from a fifth compactified dimension.

This was relatively straightforward because it was classical. The real issue with constructing a theory of everything is making these theories compatible with quantum mechanics. We've done this for electromagnetism, yielding quantum electrodynamics, but the task is much harder for gravity.

Gravitational waves are a purely classical phenomenon and they don't tell us about quantum gravity. Their detection does have the side benefit of making us slightly more confident in general relativity, though we were already very confident in its truth. As existing answers have said, the main benefit of LIGO will be to astronomers and cosmologists.

It seems to me that means gravitational waves move at the speed of light. And it seems this implies that a similar theory could be drawn up for gravity as Maxwell did for the electric and magnetic forces.

This is kind of on the right track, but a bit misguided. It's not surprising that both gravitational waves and electromagnetic waves move at the same speed, because their corresponding theories (general relativity, electromagnetism) are built on top of special relativity, which tells us spacetime comes with a fundamental speed.

This does suggest that we might be able to combine the two without much trouble, and indeed that's been done long ago. Classical gravity with electromagnetism is simply described by the action $$S = \int d^4x \sqrt{-g} \, \left( \frac{R}{16 \pi} - \frac14 F^{\mu\nu} F_{\mu\nu} \right).$$ More exotically one could consider a Kaluza-Klein theory, where you don't even put in the electromagnetic field separately, but get it automatically from a fifth compactified dimension. Or, if you restricted yourself to weak gravitational fields, you could get gravitoelectromagnetism, an approximate theory of gravity that looks almost exactly like electromagnetism.

The common thread is that these theories are classical. The real issue with constructing a theory of everything is making them compatible with quantum mechanics. We've done this for electromagnetism, yielding quantum electrodynamics, but the task is much harder for gravity.

Gravitational waves are a purely classical phenomenon and they don't tell us about quantum gravity. Their detection does have the side benefit of making us slightly more confident in general relativity, though we were already very confident in its truth. As existing answers have said, the main benefit of LIGO will be to astronomers and cosmologists.