In Lucas canon (e.g. no EU/Legends, no new Disney content), what exactly did the Sith want revenge for?
What was the specific thing that they tried to avenge?
In answer to your question, the Sith used to rule the galaxy before the Republic (and the Jedi Order) overthrew them. This act is what the Sith are taking revenge for in Star Wars: Revenge of the Sith
“The Republic has fallen, Padmé,” Obi-Wan said gravely. “The Jedi Order is no more.” “I know.” Padmé gazed at him, seeing the new lines in his face. “It’s hard to believe.” She took a deep breath. “But the Senate is still intact. There is some hope.”
“No, Padmé,” Obi-Wan said sadly. “It’s over. The Sith now rule the galaxy, as they did before the Republic.”
Padmé stared. “The Sith?” It was Palpatine who was in charge of the Repub — of the Empire. Surely Obi-Wan didn’t mean that Palpatine was a Sith Lord!
The clone wars and Order 66 are the "revenge of the Sith"
[The] Clone Wars have always been, in and of themselves, from their very inception, the revenge of the Sith.
They were irresistible bait. They took place in remote locations, on planets that belonged, primarily, to “somebody else.” They were fought by expendable proxies. And they were constructed as a win-win situation.
The Clone Wars were the perfect Jedi trap.
A small hologram of the first of the clone commanders sprang up before him. “Yes, my lord?”
“The time has come,” Darth Sidious said, savoring the words. After a thousand years, the time for revenge has come at last. “Execute Order Sixty-six.”
Mace Windu gives us this insight. As you can see, the fall of Darth Bane (and by extension
all almost all of the Sith) was clearly a turning point in Jedi/Sith relations:
“Since the fall of Darth Bane more than a millennium ago, there have been hundreds of thousands of Jedi-hundreds of thousands of Jedi feeding the light with each work of their hands, with each breath, with every beat of their hearts, bringing justice, building civil society, radiating peace, acting out of selfless love for all living things-and in all these thousand years, there have been only two Sith at any time.
To back this up, you have quotes from Palpatine that the Sith used to rule the galaxy;
and Windu stating that the Sith used to oppress people
The Sith always craved for power. In the Sith viewpoint, the Jedi have usurped the rule of the galaxy (which they did not), and that was unjust. Thus, claiming absolute power over the known galaxy by converting the Republic to the Empire and by eliminating all (almost all) Jedi, the Sith had their revenge for centuries of hiding underground.
The following is from the EU:
There are some graphic novels that are called Tales of the Jedi. In those novels one is called the Golden Age of the Sith.
Before the Republic there was the Sith Empire, the Republic and the Sith Empire went to war, resulting in the Sith being banished to the far corners of the galaxy. In the centuries that followed the Republic forgot about the Sith and got on with creating the galaxy, until a brother and sister who were hyperspace explorers came across them whilst they were checking hyperspace routes which resulted in another war. This lead to another Tales of the Jedi graphic novel: The Sith War.
Centuries later another Jedi fell to the dark side when he came across Sith teachings and started to follow them and also had other Jedi join him, his name was Exar Kun and it took the combined efforts of 1000 Jedi to defeat him. One of the Jedi who joined him was called Ulic Qel-Droma who turned to the dark side when his teacher Master Arca was killed and in his anger during a lightsaber duel he killed his brother Cay, only when he saw what he had done he broke down in grief. Nomi Sunrider, another Jedi knight, saw what happened to her friend by the man she loved; she imprisoned Ulic in a wall of light by using a Jedi force blocking technique that Master Odan-Urr had taught her. For the rest of his life Ulic couldn't use the force.
Exar kun came across a primitive race and by using the great temples on the 4th moon of Yavin to channel their life force he was able to save his spirit which stayed on Yavin 4 until Luke Skywalker established his jedi academy there, in the Jedi Academy trilogy of books (Jedi Search, Dark Apprentice, and Champions of the Force).
The spirit of Exar Kun turned young student Kyp Durron to the dark side. Durron had been a slave on the spice mines of Kessel until he was rescued by Han Solo, using an experimental Imperial Fighter called the sun crusher went on a rampage destroying imperial systems by launching resonance torpedoes into the systems suns until his friend Han Solo went after him in the Millenium Falcon which resulted in the Falcon nearly being destroyed. Exar Kun was finally destroyed by all of the students joining together and using their combined light side powers.