Bucky Barnes' arc during The Falcon and the Winter Soldier was to overcome his Winter Soldier persona. The first step was to rid himself of Hydra's mind control, which he did with the help of the Wakandans. He then started making amends, although the mistakes of his past kept haunting him.
By the show's end, Bucky has come a long way, and he's certainly closer than ever to the happiness he deserves. However, there are still many loose threads in his life that he needs to deal with before he can finally forgive himself and start fresh. The show gives audiences hope that he'll eventually get there, though, in film or a potential second season.
Still Trying: He Hasn't Forgiven Himself Entirely
TFatWS makes it clear that Bucky's traumas are far from gone. Sure, he learned how to make amends, and he keeps actively trying to be a better person, but that doesn't mean he let go of everything he did.
Bucky is his own worst judge, and even if everyone else moves on, he will still keep going until he forgives himself. The show's ending hints that he won't return to his therapist, which might not be the best idea. After all, he still has numerous issues that could use some professional attention.
Finally Happy: He Finished His List
During the show's run, Bucky has Steve's old notebook. It includes people he helped or benefitted during his time as the Winter Soldier, and he crosses them off by bringing them to justice in a non-violent way.
The list also includes individuals he wronged or hurt, both direct and indirectly, like the father of one of his many victims. Bucky has trouble apologizing to them, and fans only get to see him doing it once. Even then, the scene rushes by, and audiences don't get any reaction from either Bucky or the victim's father. Still, Bucky did it, and he took a weight off his shoulders.
Still Trying: He Still Doesn't Trust Others
Bucky has a hard time letting people in. He even found it difficult to trust Steve again, and they used to be childhood friends. Bucky still seems to fear being intimate with others, preventing him from having lasting friendships, let alone a romantic relationship.
By the show's end, he seems ready to accept Sam as not only a companion and ally but a true friend. He also holds the Wankandans in great esteem and would fight alongside them, no questions asked. However, and at least so far, Sam seems to be his only true friend left.
Finally Happy: He Owned Up To His Mistakes
Bucky spent a lot of time running away from his past. This prevented him from confronting his mistakes, and it didn't help that everyone else, Steve included, made excuses for his previous behavior.
Yes, Bucky was brainwashed when he did all those horrible things, but he still did them. Denying them won't do anyone any good. TFatWS makes it clear that Bucky no longer hides from his mistakes. He still struggles with them, but that only makes him human. And realizing that his past doesn't define him and that he can do a lot of good with his future makes all the difference.
Still Trying: He Still Feels Like He Doesn't Belong
Like Steve, Bucky feels out of place in the modern world. None got the chance to pursue a real life, though, as they both got dragged back to the soldier lifestyle. Steve's feelings of discomfort in the modern world eventually led him to break all the rules and go back to the 1940s to live the life he left behind.
Bucky chose to stay and make a life for himself in 2023. It's arguably the more mature choice, but dealing with a new reality without the one person that can empathize further alienates Bucky from everyone else. Even so, by the show's end, Bucky is actually making an effort to fit in, which is already a huge step.
Finally Happy: He Let Go Of His Anger Against Zemo
Ever since the events of Captain America: Civil War, Bucky held a grudge against Baron Helmut Zemo. After all, Zemo was the reason why Bucky began to be hunted again. The Baron killed King T'Chaka and then framed Bucky for the crime, resulting in the escalation of events that led to the Civil War.
Zemo had his reasons, and while Bucky didn't excuse them, he could at least understand them. Their stories have several parallels, but while Bucky actively fights against his worst instincts, Zemo indulges them. This is the main reason why Bucky can let go of his grudge and make peace with Zemo.
Still Trying: People Still See Him As The Winter Soldier
Reputations are feeble. They're difficult to create and easily destroyed. Bucky knows this better than anyone because no matter how much good he does, he's still the Winter Soldier in the eyes of many. The Wakandans seem to be the exception because they saw his true self and even gave him a new moniker that better fits his personality, White Wolf.
Bucky's actions as the Winter Soldier will likely never be forgotten by the world as a whole. However, he can use his time to atone for the crimes of his past. Like Natasha, he too can wipe the red from his ledger.
Finally Happy: He Reconciled With Steve's Departure
Avengers: Endgame hinted that Steve told Bucky about his plan to return to 1940. Alas, the scene isn't part of the movie, and fans can only guess how the conversation went. During their joint therapy session, Bucky unintentionally reveals to Sam that he still has some unresolved issues about Steve's departure.
Steve and Bucky's relationship is the subject of many fan theories. Many people ship them and believe they should've ended up together, while others insist they're just friends. The MCU will probably never dig deeper into their complex dynamic, but their relationship remains fundamental to both characters. At the end of the show, Bucky comes to terms with Steve's absence, reiterating his support for Sam's new identity as Captain America.
Still Trying: He's Still Fighting A War
During his therapy session, Bucky reveals that he feels trapped in a cycle of war. He reveals that the only time he had for himself was in Wakanda. Other than that, he spent his life going from battle to battle without any real break.
As a superhero, this won't ever change. Bucky will spend the rest of his life fighting, whether it's against "the big three," or rogue agents, or corrupt governments. Even if he chooses to step away, his commitment to justice and his desire to atone will eventually lead him to step back into the fight.
Finally Happy: He Has A Support System
TFatWS ends with Sam and Bucky at a party. After their battle, and with a job well done, they can momentarily relax and enjoy, even if just for the afternoon. Surrounded by Sam's friends and family, Bucky actually integrates and socializes, making an effort to fit in.
It's a wholesome and inspiring scene because it shows that Bucky is no longer alone. He has Sam, but most importantly, an entire support system that he can rely on when things get rough. For so long, Bucky was alone with himself and his demons, so it's nice to see him in the company of people who accept him for who he is and encourage him to be better.