Syl finding a rare plant in the middle of a highstorm and bringing it back to Kaladin just because she'd seen him with one earlier, but he'd lost it. She did this unprompted (despite being none-to-bright up until this point), just because she didn't want him to give up hope. This would be heartwarming enough if she were a human, but, up until this point, spren weren't even corporeal. Also a Funny Moment; see that page for details.
Syl with a tiny sword, fighting off the deathspren as they come to claim Kaladin after he was strung up in a highstorm.
When the men of Bridge Four give Rock a shaving razor.
When Kaladin needed four more people to bait the Parshendi with him, everyone in Bridge Four volunteered. Kaladin then reflects on how much they have changed. In the beginning, they would have shoved the newest and the weakest to the front in order to be an easy target. Now, they all would willingly go into danger so long as Kaladin is there.
In general, Bridge Four is full of heartwarming moments. They are so loyal to Kaladin that they would work themselves to the bone so that they wouldn't disappoint him.
Dalinar giving up his Shardblade to save Sadeas' bridgemen. It may have saved Kaladin's faith in humanity in general and the lighteyes in particular.
Dalinar: What is a man's life worth? Kaladin: The slavemasters say one is worth about two emerald broams. Dalinar: And what do you say? Kaladin: A man's life is priceless. Dalinar: Coincidentally, that is the exact value of a Shardblade. So today, you and your men sacrificed to buy me twenty-six hundred priceless lives. And all I had to repay you with was a single priceless sword. I call that a bargain.
About halfway through the book, Dalinar decides that although Renarin is too sickly for battle normally, that wouldn't matter with Shardplate, and promises to win him a set. At the end, after he's given up his Shardblade, he realizes that it's easier to just give him his own plate.
A very subtle one, but about halfway through the book we see the King's Wit (essentially a professional Deadpan Snarker) sitting at the entrance to a feast insulting everyone as they show up. We see him give clever insults to about twenty different nobles, but he just says "Hi" to Dalinar.
It's pretty clear that despite what Wit says, there are people he genuinely likes and respects, and people he doesn't. Dalinar's entire family fall into the first category, only ever getting good-natured ribbing on Adolin's part. Wit actually seems to use Renarin as a straight man for many of his jokes. Sadeas, however, gets the vulgar humor.