Star Trek: Discovery, like some of the other Star Trek series, got off to a rather rocky start which continued throughout the entirety of its first season. It was, in fact, the season that many Star Trek fans say they couldn't get through which made them stop watching the series. While I did watch the entire first season, I agree there were some rough spots that, had they been fixed or eliminated altogether, would have made Discovery much better in its first season.
The first would be the death of Captain Georgiou because of Michael Burnham's actions. Having the captain killed could have had a much more powerful effect if it hadn't been because of a choice Burnham made. There was still a way to have the captain die and take Burnham out of action for a while, if the writers and producers had chosen that route. What about a battle that killed the captain and gravely injured Burnham? Or left Burnham with memory loss? There were so many other viable options.
The second would be, of course, the Klingons. Everyone hated the look of the Klingons, and it's not that they shouldn't have been included, but they should have looked like the ones from Star Trek: The Next Generation. There was no reason to change their appearance.
The third would be Michael Burnham's mutiny. Though this ties in with the first change, this one was important in that, when Burnham returned, no one trusted her. She had to work to earn their trust and make her way to the captain's chair. Placing that kind of darkness on a main character didn't set the right tone with fans, and some tuned out because of it.
Making the Klingons the villains. While Star Trek: The Original Series had some episodes where the Klingons were the villains, they didn't run an entire season based on that premise. They varied the enemies. Of course, this goes back to Discovery being serialized and not episodic. I think, though, if the producers had chosen the episodic path, Discovery would have had a much better first season.
And finally, making Captain Lorca a villain. Many of the crew that had lost their captain aboard the USS Shenzhou transferred to Discovery. Not only did their captain die in battle, but the captain's first officer betrayed everyone and was carted away to prison. Then the ship was captained by Gabriel Lorca who turned out to be the Mirror Universe Lorca. So it's no wonder the crew had trust issues in the entire first season.
An additional change I would have liked to have seen would have been the setting. Instead of making it a prequel, Discovery could have been set in the timeline after Star Trek: Voyager and/or even in another Quadrant. Only they could have been assigned there purposefully. They would have been able to encounter new aliens, new villians, and had some amazing episodes without having to utilize the captain as the enemy.
Overall, personally, I think the series would have been better off having Burnahm in the captain's chair from the beginning. She was a strong character, much like Kate Mulgrew's Captain Janeway, and there were plenty of other ways to have created conflict between her and the crew without making her the first enemy.