Incoming wall of text! Sorry, I like to think through interesting and unusual scenarios like this. Hope this lot helps!
First up, I'd recommend reading
Moonfall (Jack McDevitt, 1998). The premise: a comet smacks into the moon and shatters it; it's done pretty well, with scientifically-plausible results.
Book recommendation aside, time for some science.
First up, scale your destruction. Actually
destroying the moon... yeah, totally not going to happen. Full nothing-left-but-smoke destruction would take mind-bogglingly huge amounts of energy. We're talking antimatter by the truckload. Blowing it up so hard that the remaining bits disappear into space takes less energy, but a ridiculously large amount. Lifting things completely out of Earth's gravity well and off into interplanetary space takes so much energy that we've never managed it with anything bigger than a small van or truck, so with the millions and billions of tonnes of rock in the moon, you're still looking at crazy amounts of energy to lift even a few fragments.
So, what can we do? Well, a comet coming in from the outer solar system would have a LOT of energy. Easily equivalent to a few dozen tonnes of antimatter. That could, potentially, break the moon into chunks and send those chunks moving apart at an appreciable speed. As I said, there's at least one book out there with this premise.
At this point, your biggest problem is actually the smaller chunks. The biggest chunks have enough mass to have their own gravitational pull, so you'd probably see them moving apart a little way, and then falling back towards each other. Given time (and I mean astronomical time here, so look at a couple million years) some sort of moon would reform from the pieces, slightly smaller and perhaps a little lumpy, but a single body once more. The smaller bits are more dangerous because being less massive means they're easier to push around, and the impact that shatters the moon is going to send some of them out at sufficient speed to break away from the moon's gravity - remember, just because it's in bits doesn't mean the gravity goes away; you've still got the same amount of mass in the same rough space, so things would be able to orbit the cluster of fragments just like they did when it was whole. From far enough away, you wouldn't notice much difference, gravitationally. At least, not at first.
Those smaller bits that break away would curl round, following something close to the moon's original orbit, but plus or minus a bit, depending on which direction they left the broken moon, and how fast. That plus or minus might be significant; if they exit "backwards" (opposite the moon's direction of travel) then they might be slowed enough to fall out of orbit and hit Earth. If they're ejected "forwards", then their orbits might become very elliptical, meaning they rise above where the moon used to be, to then fall back in towards Earth later - and if they're big or moving at speed, there's a decent chance of them hitting or gravitationally affecting other chunks of moon on the way, making it hard for scientists to predict what will happen and which rocks will end up where. Some of the bits that do make it to Earth could be big enough to pose serious threats. They may (or may not) fall somewhat slower than meteorites from outer space by virtue of their having started at the moon rather than just intercepting from somewhere across space, but if they're big enough then they'll still pack a punch.
The first rocks to fall will be small. Being smaller means they're more easily accelerated, so they'll deviate the most from the moon's orbit and therefore cross the distance soonest. Look at the bit in Armageddon where the BIG rock is preceded by little ones that pepper some cities. We're knocking these chunks off a moon-sized rock, so "small" is a relative term here; these could be anywhere from pebbles that burn up in the upper atmosphere and make a pretty light show, all the way up to house-sized rocks that obliterate whole neighbourhoods. With the largest variation in trajectory, these will be first to start falling and the last to stop.
Next will be the city-smashers. Rocks the size of tower blocks; these will be a shade slower, but much heavier. These will likely be big enough that there will be warnings about them; radar can track smaller than this, but the little ones will be hard to differentiate in the cloud (and there will be so many that there's little point warning for them all). These will stand out a little, so they might be worth tracking. These will stop falling before the little rocks.
In the thickest part of the rain of meteorites will come the real behemoths. Depending on how hard the moon was hit, the upper limit for the size of these rocks could vary from "hundreds of meters", past "rock the size of Everest", all the way up to "significant portion of what was once the moon". Most likely though, for an impact with a roughly average comet, anything bigger than a city block or three will be massive enough to fall back into the cluster. These will only feature in the worst part of the rockfall, probably; being so massive means that it takes a lot of energy to change their orbit, so they'll be clustered closer to the Moon's old orbital path. There might be a few that are borderline cases, with limited observations by scientists (who kinda have their hands full under difficult conditions) making it hard to tell whether they've escaped the gravity of the used-to-be-the-moon cluster and will pose a threat, or whether they're just teetering on the brink a bit before falling back into the main group to get started on forming a new moon.
If the moon is broken into small enough bits, Earth may end up with rings. If not, a few chunks of moon might get stranded at the
Lagrange points, particularly L4 and L5. This
happened to Jupiter at some point, but it's less likely here. Our L-points are pretty small and regularly perturbed by things (like Jupiter). It's still possible though, especially for the little chunks. The most likely outcome, however, is that we'll end up with a slightly smaller reformed moon, and the rest will either fall to Earth, or orbit around for a while before being gravitationally hoovered up by the "new" moon or the Earth. Space travel would be even more dangerous for a while, with all the new rocks flying around providing
lots of things to hit, or be hit by. Amusingly enough, the moon
would end up decimated, in the orignal sense: reduced by a tenth, or some smallish fraction at least.
As for your tidal and tectonic effects... yeah, there would be some of that. The most pronounced effect would be in the initial phase, as the largest chunks moved apart, moving the Earth/moon system's centre of mass closer to the Earth, reducing the moon's effects on tides and tectonics. It might also have implications for other things - the moon moderates angular momentum, acting in a similar way to the big weighted ring on spinning tops, so if the mass concentration is affected by a large enough degree, Earth's stable 23(ish) degree tilt might change as the Earth wobbles like a spinning top does when it slows down. That could shift the tropics, which would affect weather and all sorts. If it shifted far enough, the "months of day/night" thing that happens in the Arctic and Antarctic circles would become more widespread, with all sorts of implications for the ice caps, and for crops, and general wellbeing of people.
Whatever happened, chances are the effects would stabilise as the moon reformed, but the lower mass of the reformed moon would mean that the moon's effects on Earth would be reduced somewhat.