Are we closer to a theory of everything thanks to the detection of gravitational waves? A couple of weeks ago I heard an astronomer explain that one of the latest detections of gravitational waves was accompanied by simultaneous detections of the same astronomical event in various other wavelengths.
It seems to me that means gravitational waves move at the speed of light. And it seems this implies a similar theory could be drawn up for gravity as Maxwell did for the electric and magnetic forces. Am I on the right track?  
Does this mean that a theory of everything is closer thanks to the discovery of gravitational waves?
 A: 
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.
A: The "similar theory" in this context is general relativity. You get gravity waves as classical predictions. And so far that's what is being observed. It's the gravity equivalent of somebody waving a really big magnet back and forth at a few 100 Hz. 
It's really cool. But so far it does not take quantum mechanics to  understand.
In both cases, it would be nice to have a quantum theory. In the electromagnetism case, that's done. Thank you to a bunch of great minds in the late 19th and early 20th century, notably Feynman, and lots of other people were critical in the development. But for gravity, so far, things are illusive.
The reason that went relatively fast was that we have lots and lots and LOTS of experiments relating to E&M. At a large variety of energies, and under a huge variety of conditions. Some easy to calculate and some less so. From chemistry to solid state circuits to electric motors to radios. To "why is gold gold-colored instead of silver-colored?" To super conductivity. To electric train sets. To how does this electric gadget on my car reduce rust? It's all E&M. (Well, with some nuclear on the edges I guess.)
So, a detecting a few gravity waves is really nice. But there's still a long way to go.
A: No, I don't think so.  Gravitational waves are a prediction of General Relativity, which is now a little over a century old.  GR also predicts that gravitational waves will travel at $c$.  Gravitational waves are, however, absurdly hard to detect, and so it has taken us pretty much a hundred years to build machines sensitive enough to detect them.
So, essentially, the detection of gravitational waves are another test of GR and, now, one which it has passed.  Of course, and probably of more interest since I don't think anyone expected GR to fail this test, the observation of gravitational waves open up a whole new area of astronomy which is a very wonderful thing.
The significance of the events (which I think at this point means the single event, GW170817) being observed in the EM spectrum as well is that this means that we have seen events for which the precursors were neutron stars rather than black hole: black hole mergers are very 'quiet' in the EM spectrum because, well, black holes are black and at the point they merge they have long ago eaten any accretion disks they may have.  But neutron star mergers are not, so what we've now seen is two neutron stars merging, and we have seen this event both as gravitational waves and electromagnetically.  This is good at least because it means that LIGO can see these less extreme events, and because it gives us additional confirmation that gravitational waves travel at $c$ (if they did not then GR would be wrong).  Also it is good for observational astronomy of course!
