This is very open-ended since many details have been left unspecified (e.g. how the planet is destroyed, the objects which are to be affected, the time scale of interest, ...), but here are some possibilities.

If the planet is broken up into a bunch of pieces, then those pieces would begin to drift around in the solar system. They may form an asteroid belt, like what our solar system has between Mars and Jupiter. Alternatively, the pieces could rain down on other planets, causing meteor showers (meteor showers are mostly caused by comets, but the pieces of the planet in this instance would be analogous to the pieces which come off of comets when they approach the sun) or a more serious impact event.

One can also imagine simply turning all of the mass of the planet into energy. For an Earth-mass planet, this works out to ~5.4×10⁴¹ J. ("J" means Joule, which is a unit of energy) For comparison, the sun puts out 383×10²⁴ J per second (the luminosity measurement on that page). Thus, the energy from the destruction of that planet is about what the sun puts out in about 44 million years. This seems like a lot, but supernovae can release 10⁴⁴ J, 1000× more than our exploding planet.

Considering how hard it is to spot supernovae, I think it is safe to say that anyone in the same galactic neighborhood might take notice, but observers on the far side of the galaxy would have to be looking in exactly the right place at the right time to even see it.

Removing or dispersing the matter that made up the planet would also change the gravity acting on the other objects in the solar system. This means that there would be changes to their orbits, tides, etc.

Jupiter is known to affect trajectories and impact rates of cosmic bodies on the other members of our solar system, so removing a planet like that could lead to changes in the impacts experienced by the remaining planets.

There would certainly be other effects, but it is impossible to explore all of them here.

The other planets in the solar system would move to different places, because they wouldn't have the gravity of the big planet to keep them where they are not. Pieces of the big planet would fly out into space, and some of them would land on other planets.

That depends on what you mean by "destroyed". If you mean "removed from existence entirely", then the fact that the planet no longer is exerting gravity would have an influence on the behavior of the other planets in its star system. If you just mean "benuded of all life", then the planet would continue on its orbit and other planets would feel its gravity as normal.

Well, first the debris might cause damage to surrounding planets or moons. In time, however, the leftover debris would probably begin to orbit the sun. Any moons the planet had would have their orbit disrupted. If the planet were just transformed into a roughly planet-shaped cloud of rock chunks, the moon might be able to continue orbiting the planet-chunks. If the planet were totally blown up and all the chunks went everywhere the moon would probably end up orbiting the sun, or perhaps being pulled into the orbit around a nearby planet. It depends on what planet, when, and how it happens!