How could they achieve this ?
The gravitational binding energy of the Earth is about 2.9 x 1031 joules. You pump that amount of energy into the Earth somehow, you'll reduce it to a ring of gravel orbitting the sun. To blow the bits out of the Solar System entirely, you'll need about 10 times more energy again. That's a lot of energy... about 7 x 1015 megatonnes TNT equivalent.
Does humanity currently have the means to do this relatively quickly? For example are all the nukes in the world sufficient for this ?
Haha, no. This is so many orders of magnitude more power than all the worlds nuclear weapons ever it isn't even worth thinking about. It is more power than was delivered by the Chixulub impact. You're asking for more energy than the Theia impact hypothesis that would have resurfaced the entire earth and formed the moon. Let me just repeat that last bit for you: crashing a small planet into the Earth wouldn't be enough energy to do what you want. Complete annihilation of 161 trillion tonnes of antimatter with an equal volume of matter would release enough energy, but you'd have to do it in the very centre of the Earth in order to ensure there was no wastage.
All humanity's knowledge and resources are at their disposal and they have access to all kinds of weapons currently present in the whole world
They'd know well enough that they couldn't do it. At all. Not even close.
Assuming this is possible, in what cases would the Moon and this group of people would be left intact/unharmed ?
If you blew up the Earth, the Moon is practically at ground zero. It'll be hit by a lot of stuff moving pretty fast. Sitting on the far side of the moon is unlikely to be safe enough... secondary impacts caused by ejecta from impacts on the Earth-facing side of the moon, and from slower-moving debris from Earth itself will still likely hit you. You'd have to be buried very deep. I'm not sure how deep. With a violent enough explosion (like, if you wanted to blow all the bits of Earth out of the Solar System, not really reduce it to a ring system) would risk destroying the moon, too.
What happens to the Moon after that ? Would it drift through space and eventually orbit another planet in the solar system ?
If you clapped your hands and vanished the Earth, the moon would continue to orbit the Sun with pretty much the same year length as Earth has now. A violent end to the Earth might knock the moon into a slightly different orbit, but it is unlikely to throw it into some completely different orbit or out of the Solar system entirely.
The result can only be considered as "successful" if and only if there is a nice big absolute nothing left where our lovely little blue planet used to be.
Consider moving the Moon to the Earth-Sun L3 point, which is diametrically opposite the Earth on the far side of the Sun. Get your deluded planet-demolishers really, really drunk, and have them wake up there. Tell them they were successful. They're probably too daft to realise what happened.
This will also be far, far too hard to do with any plausible current or future technology, but its a lot easier than disappearing the Earth, that's for sure.