Law is a mysterious character: even after spending so much time with him, his true goals are unclear. We know he wants to reach Laugh Tale, and uncover the true history, but, we don't know why.
Flevance, the town the World Government lied to and destroyed:
However, his flashbacks gives us clues to his other goals.. Flevance, White Town, Law's hometown, was destroyed due to the White Lead trade. Despite knowing how toxic it was, the royalty of Flevance continued to mine and sell it, while keeping the information abouts its toxicity from the people of Flevance and the rest of the world.
This means that the World Government and Royalty willingly doomed Law's family and home to death. All the suffering he went through, watching his family be murdered, watching his home be burned to the ground, surviving by hiding out in a cart of corpses... all of it was because of the World Government, and the Royalty who ruled Flevance.
However, those Royals did not share Flevance's fate: they escape with the WG's aid.
So, the family that sold Law's town out to die in order to sell White Lead, is still alive, and most likely at the Holy Land, Mary Geoise, considering who helped them.
This is Law's true revenge. Law's goal is to destroy the Royal Family that doomed Flevance town to destruction and despair.
The Second Survivor of Flevance?
However... I think there is hope. It's always been said that Law was the sole survivor of Flevance: however, I think Lammy, his sister, may have also been saved.
While Oda shows off Law's dead parents before this, and the corpses of his friends, we have no direct proof Lammy died. Doflamingo even asks Law if there were any other survivors, foreshadowing the potential that someone else got out.
But, who could have saved her? Who do we know who saved people left to burn alive by the World Government, who is known for rescuing children, and who fights against the WG's injustice?
"But CoC, Lammy was already dying from White Lead poisoning?!? She would have died anyway, even if she did escape Flevance!!!"
Don't worry, I know a crazy pervert who can heal Lammy with MAXIMUM CANDY HORMONE OKAMA POWER!!!
This is Law's last hope for a happy ending, discovering that his sister was saved by Dragon and the Revolutionaries all those years ago, when they came to try to rescue the remaining citizens, but could only save one: Lammy.
"See, Law? There's no need for despair... There is always a merciful hand of salvation ready to help." (RIP Flevance Nun, you were a sweet-heart)
If that merciful hand of salvation was Lammy being saved by Dragon, then this connects Law further to the Revolutionaries: I wouldn't be surprised if Law meets up with Monkey D. Dragon soon.