When Game of Thrones began, Robert Baratheon was sitting on the Iron Throne and House Baratheon was in good standing. His death resulted in a fight over the throne that involved Joffrey Baratheon being named king, despite not being a true Baratheon, and Robert's brothers, Stannis and Renly, each naming themselves as the true heirs.
In the end, these would-be kings all died and House Baratheon dwindled almost out of existence. How could such a powerful house decline so quickly? There were many mistakes made by the Baratheons along the way and many things they could have done better to win the game of thrones.
Robert Marrying Someone Else
In order to win the Iron Throne in the first place, Robert had to make some deals with people he didn't necessarily like. This included Tywin Lannister, who betrayed King Aerys Targaryen in exchange for Robert marrying his daughter, Cersei.
It was an unhappy marriage and one that put the Lannisters in a plotting position. However, in the books, there is a plot revealed to have Robert marrying Margaery Tyrell. Had this happened, Robert would have maintained wealth and power while ridding himself of the Lannister influence.
Renly Teaming With Robb
In the aftermath of Robert's death, many different kings declare themselves, including Ned Stark's eldest son, Robb. Unlike the others, Robb had no desire to sit on the Iron Throne but rather wanted revenge against the Lannisters and freedom for the North.
Robb sends his mother Catelyn to meet with Renly Baratheon to discuss a potential alliance. While Renly would have had to allow the North to separate from the other kingdoms, the alliance would have made for an unbeatable army. Unfortunately, Renly was killed before such a deal was made.
Stannis Being More Patient
Despite his unpopularity and stern demeanor, Stannis might have been the Baratheon who was best suited to be king. He didn't much want to job, but since he was Robert's true heir, he felt it was his duty to take the Iron Throne.
But while Stannis believed in justice and doing what was best for Westeros, he was also impatient in achieving what he thought he was owed. This led to his ill-advised march on Winterfell to fight the Boltons. Had Stannis waited, he could have gained support from other Northern lords and avoided his own downfall.
Ned Endorsing Renly
When it became clear that Robert was going to die, everyone started making their own power grab. With Stannis away from King's Landing, Renly took his chance to name himself as the rightful heir and tried to convince Ned Stark to back him.
Littlefinger also felt it was wise to bet on Renly, but Ned knew that Stannis was the true heir and refused to budge. Had Ned gone along with this plan, he and Renly would have been able to overthrow the Lannisters, along with Littlefinger's help, and take King's Landing easily.
Stannis Not Listening To Melisandre
The relationship between Stannis and Melisandre was a difficult one. Her dark powers certainly helped him gain some big victories, but his reliance on her also may have been his biggest fault. She was able to convince him that he was destined for the throne, which seemed to cloud his judgment.
Also, despite being a fair man, h followed through with her burning people alive, which didn't win him many allies in Westeros. This also led to Stannis sacrificing his own daughter, and was clearly the moment he lost the war.
Robert Surrounding Himself With Allies
Even though it was Robert who was sitting on the Iron Throne, it was widely known that it was actually the Lannisters who held all the power in Westeros. And this is largely Robert's own fault for letting them surround him at all times.
Along with being married to Cersei, Robert also allowed the kingdom to be indebted to Tywin, for Jaime Lannister to be in the Kingsguard, and for Lancel Lannister to be his squire. If Robert surrounded himself with more people he could trust, the Lannisters wouldn't have been able to control him.
Stannis And Renly Teaming Up
The Baratheon brothers were certainly not a close-knit family. In fact, each of them admits separately that they don't love their brothers. This disdain between them may have been their downfall, especially in the case of Stannis and Renly.
Once Robert died, the two remaining Baratheon brothers split the support for their family. While both men acted immaturely, Stannis was the rightful heir and he even offered to name Renly as his heir if he joined him. If Renly had just admitted that he wasn't fit to be king, a lot of trouble could have been avoided.
Jon Snow Became Jon Stark
The moment that showed how Stannis could be a good ruler was when he brought his army to the Wall to help the Night's Watch defend it from the Wildlings. This also led to a relationship of mutual respect between Stannis and Jon Snow.
Once again, despite Stannis's cold demeanor, he offered a pretty generous proposition. In exchange for helping win the Northern lords to his cause, Stannis would legitimize Jon, making him Jon Stark, and naming him as Warden of the North. This would likely have won Stannis a lot of support in the North and allowed him to defeat the Boltons.
Robert Being A Better King
Robert was known as a great warrior who led a rebellion that overthrew the Targaryens and won himself the Iron Throne. As it turns out, taking the throne in battle does not guarantee a good king as Robert proved to be quite incompetent.
He spent his time drinking and visiting brothels, uninterested in attending council meetings or learning how to rule the kingdom. As a result, Robert made a lot of enemies who felt things would be better off if he wasn't king. Had he taken the job more seriously, there might have been peace.
Ned Reveals The Truth
Ned Stark may have been the most honorable man in the show, but he also made a lot of dumb mistakes in the name of honor. His biggest mistake came after he discovered that Robert's children were actually the children of Cersei and her brother Jaime.
Instead of going to Robert with this news that would have pretty much ended House Lannister, Ned decided to tell Cersei in order to give her a chance to escape King's Landing with her children. His act of mercy ended up getting Robert killed along with Ned himself.