I assume you mean these three as applied to people. "Integrity", for example, means something different (but metaphorically related) when applied to something like a building.
Honesty generally refers to truthfulness, someone who tells the truth and is free from deceit.
As a boy, John's honesty was without question: although no one saw him damage the neighbor's yard, he came forward and told his parents that he was the one who did it.
Integrity generally refers to soundness of moral character, including such things as honesty, reliability, decency, and sincerity. Like a strong building, someone with integrity can be relied on to say what they mean, to do what they say, and to know right from wrong.
Even though her colleagues were making enormous amounts of money, she had too much integrity to sell a product she knew was too much of a financial risk for her customers.
Veracity generally refers to accuracy and correctness. It it generally applied to things or events, but when applied to people it is synonymous with "honesty".
The veracity of a story does not depend on how many people believe it to be true. Instead, we have to look at the facts to determine what really happened.
If you specifically want to say a person is truthful, then use "honest". If you want to talk about their overall character, then use "integrity". Again, "veracity" probably is best used for objects, not people, especially since there are many other words available.
Related nouns: sincerity, probity, rectitude, truthfulness, incorruptibility, righteousness, honor, candor, trustworthiness, fidelity
Related adjectives: sincere, truthful, incorruptible, righteous, honorable, principled, virtuous, moral, trustworthy