Where would Moon go is obvious and answered by others - interesting is how will human colonies deal with the consequences?
Where would Earth debris fly, depends of the energy and direction of the death blow.
- With enough energy, debris would fly all over the inner solar system - and Moon, after surviving first impact, will continue near Earth orbit (which will be also path of most of the Earth debris). Such debris are danger for Mars and Moon (and other asteroid mines), but not Pluto.
- With less energy, debris will fly close to original Earth orbit and with just little energy, Moon might be able to orbit remnants of the Earth, and debris would be less of a danger for Mars.
Big chunk of the debris might re-accrete into new Earth - it will not be habitable for millions of years, but at least other human colonies would be safe from debris. If not, we have new (inner) asteroid belt - some chunks could cool down in few hundred/thousands years to be useful for mining for resources like metals. Good source of iron, already orbiting - and we will need it.
Absence of Earth will not have any major impact on the other planets - Sun's gravity sets the rules.
What would have impact would be flying debris. Bombardment of human colonies on Mars and especially Moon by Earth remnants would be great danger for centuries. Moon would have likely most of the colonies on the Earth side - and they would be heavily impacted by original explosion.
What about other human colonies?
Survival of the other the human colonies would depend on how independent their ecologies are. It will be rather hard, I presume, to grow all the food, but food production would be obviously highest priority. Quality of life would decrease substantially on Mars - but it's better to struggle than to be dead. If Moon is just tourist trap, most or all population would quickly starve.
Pluto is so far that it has to be independent (nuclear-powered hydroponics anyone?) - so Pluto most likely will survive, even if Mars could not survive bombardment by Earth debris and damage to its ecology. Any rogue debris from Earth will take long time to get to Pluto, and it's trajectory could be simply changed to less dangerous. So maybe it will be Pluto which will repopulate Solar system.
If Earth debris form another asteroid belt, survivors from Pluto and Mars would have excellent mining base to build interstellar spaceships.
Terraforming Mars is IMHO fools errands, Mars would lose new atmosphere for exactly same reasons it lost old one, especially with the extremely limited resources on Mars. Of course it depends of danger of bombardment by Earth debris, and how close is terraforming from being finished. Maybe if some chunks of frozen Earth water can be safely dropped on Mars, destruction of the Earth could even speed up terraforming Mars - not sure what plans has OP, it depends on the storyline.
Possibly, scarce resources on Mars would be better spend to build underground cities on Mars (protection from bombardment of Earth debris), and developing interstellar travel.
It will be most important to settle some asteroids and get them moving toward other solar systems: whatever destroyed Earth, may return and destroy rest of the human colonies. You definitely want to get out of there. This Solar system is doomed, dude.