As the answer to this question ponders, Dumbledore seems to have left a lot of information out when giving Harry, Ron and Hermione their Horcrux assignment. Let's consider the following points.
Harry didn't know how to destroy Horcruxes.
"I wonder how Dumbledore destroyed the ring?" said Harry. "Why didn't I ask him? I never really..."
His voice trailed away: he was thinking of all the things he should have asked Dumbledore, and of how, since the Headmaster had died, it seemed to Harry that he had wasted so many opportunities, when Dumbledore had been alive, to find out more...to find out everything...
(Harry Potter and the Deathly Hallows, Chapter 6, "The Ghoul in Pyjamas")
Dumbledore made no plans to inform Harry of how to destroy Horcruxes.
It was only sheer, dumb luck/quick thinking on Hermione's part that meant that the trio knew the theory of Horcrux-destruction. Dumbledore just left the Hogwarts library books lying around in his office and made no attempt to bequeath them to Harry. At the start of their quest they had no practical means of destroying them.
Harry felt hurt and betrayed that Dumbledore knew but didn't tell him.
He could not hide it from himself: Ron had been right. Dumbledore had left him with virtually nothing. They had discovered one Horcrux, but they had no means of destroying it: the others were as unattainable as they had ever been.
(Harry Potter and the Deathly Hallows, Chapter 16, "Godric's Hollow")
Dumbledore had no reason not to tell Harry.
Knowing how to destroy Horcruxes is clearly an integral part of any mission to destroy Horcruxes! With the Hallows, Dumbledore later says that he had intended to be cryptic in an attempt to let Hermione's natural caution prevent Harry from going too power-hungry. This is not the case with the Horcruxes.
Dumbledore knew his official will would be subject to Ministry of Magic intrusion.
Regardless of whether the Ministry's motives were kind or cruel Dumbledore clearly knew that the bylaw about the Ministry having the right to search will items could provide difficulties in terms of giving things to the trio. This is evident from the care he took to hide the true meaning of Beedle's book and the elaborate magic surrounding the Snitch he gave Harry. Dumbledore must've known that the Ministry may not allow Harry to receive the Sword of Gryffindor. Trying to give it to Harry gave him a hint but this is as close as Harry gets to finding out how to destroy Horcruxes.
Dumbledore had other methods of giving things to Harry.
Even though he knew that he was dying, Dumbledore obviously took care to tailor his will to helping Harry. Yet he could've simply given things to someone he trusted at Hogwarts like McGonagall to pass onto Harry.
Dumbledore knew basilisk fangs would be supremely useful to Harry.
Dumbledore knew that, if Harry didn't get hold of the sword, basilisk fangs remained the main key tool Harry needed for Horcrux-destruction. Ron and Hermione ended up going down to the Chamber anyway so Dumbledore could've saved them the hassle.
Why didn't Dumbledore give Harry more of a head-start by telling him about the Basilisk fangs? And why didn't he put the fangs, the sword, the Beedle book, the Deluminator and the snitch in a box for McGonagall to give away after he died?
"From all that I've read, what Harry did to Riddle's diary was one of the few really foolproof ways of destroying a Horcrux."
"What, stabbing it with a Basilisk fang?" asked Harry.
"Oh, well, lucky we've got a large supply of Basilisk fangs, then," said Ron. "I was wondering what we were going to do with them."
(Harry Potter and the Deathly Hallows, Chapter 6, "The Ghoul in Pyjamas")