Detailed further in Captain America's page, but Bucky's first birthday as Captain America. It wasn't the best time for the Avengers, but everyone got together to help Bucky celebrate for the first time since WW2, with Natasha teasing him for his age by placing near enough 90 candles on his cake. Before he blows them out, Mockingbird stops him to ask if he has any wishes to make first, and he responds with, basically, nope. For the first time, he's happy with his life.
Pretty much everything about Bucky and Natasha's relationship, before their controversial breakup. The two fell in love during their time as Soviet pawns, and were so infatuated that their relationship began breaking down his programming, and the sight of what they were doing to him disgusted Natasha so much that, if not for her own brainwashing, would have likely made her defect. Decades later, they meet up first as enemies, but once they're on the same side Natasha takes it upon herself to help him settle as the new Captain America, and the two basically agree that time as KGB assassins was pretty shitty, but they always remembered their original affair as the one good moment of it, before reuniting that relationship.
To a lesser extent, his relationship with Sam, The Falcon. Sam originally doesn't see him as anything but Steve's disturbed former partner, but goes out of his way to protect Bucky in order to honour Steve's death. The two become such good friends from all of this that Bucky ends up considering him his best friend, besides Steve of course.
In the miniseries Avengers/Invaders, WII-era Bucky and the Invaders have been brought to the future after Steve's death. During the ensuing Let's You and Him Fight, Bucky throws WWII Cap his shield... and misses, so that it falls off the airborne helicarrier to the ground. Cap is sanguine about it, sure that Bucky will get it back, but Bucky is absolutely humiliated... up until the shield is thrown back to Cap again. Bucky's future self, then Captain America, apparently knew where the shield would landnote It's really, really ambiguous if Bucky and Namor, the only two Invaders who were both sent to the future and lived to see the day of their arrival, had any memory of what was going on. Namor indicated his younger self needed to learn a particular lesson and knew how to give it, while Bucky went and cryptically warned himself about his "death," but the Cosmic Cube was involved. and retrieved it for him— just like Steve knew he would.
Also in Avengers/Invaders, Bucky Cap vaguely warned WWII-era Bucky about his death, telling him only that if Steve ever told him to drop off and let a missile go, to listen to him, hoping to spare himself and Steve the whole Winter Soldier mess. WWII Bucky later got his hands on the Cosmic Cube, and remembering Bucky Cap's warning, flirted with the idea of changing reality. However, he decided that, even if Steve told him to save himself, he couldn't know that people wouldn't be hurt if he didn't let history play out as it already had. So teenage Bucky chose to go ahead and go through with his own possible death. Instead, he used the Cosmic Cube to bring his best friend Toro back to life in the future.