At the beginning of the San Diego Comic-Con 2011 Amazing Spider-Man panel, an excited "fan" rushed to the Q&A microphone with an urgent, emotional message to share with the crowd... Turns out the fan was Spidey himself, Andrew Garfield, who proceeded to give a heartfelt speech about the character. You can see it here.
Early in the film Peter tells Uncle Ben that he's "A pretty good dad."
Flash Thompson, the same guy who beat up Peter in the first 15 minutes, tries to console Peter after Uncle Ben's death. When Peter slams him against the wall, he actually has a bit of a 'get it out of your system' attitude to the whole thing. It's especially nicer at the end when the two are now friendly with each other.
Let's elaborate on this: At the start of the film, Flash is seen as a major bully who literally kicks Peter while he's down, after beating him up. Peter's first real act of abusing his new-found superpowers is to completely humiliate Flash on the basketball court, with all his peers watching. When his Uncle dies, none of the students want to talk to Peter. They all keep their distance from him, moving out of the way and just staring at him. Except Flash. He goes to talk to him and offer his condolences and sympathies when no one else will. Peter then understandably freaks out and smashes Flash up against the lockers. Does Flash get angry and fight back? No. He keeps talking to Peter, calmly and soothingly, asking him if the venting has made him feel even a little bit better, telling him how sorry he is for Peter's sake. That's real compassion.
In the final scene he hugs Peter!
Crosses over into Fridge Brilliance since by 'defeating' Flash in the basketball incident, he unknowingly established himself as the new top bully hence why no one wanted to approach him (he was already a loner and now he was a loner bully). Flash, perhaps among other things, realized that he might be the only one willing and able to even approach him. For that matter, Flash is also approaching someone that humiliated him multiple times (basketball, name incident, but certainly Peter must have known Flash well enough to know his real name and his embarrassment over it). Flash doesn't just demonstrate compassion, he demonstrates courage.
Peter tries to reason with Dr. Connors during the fight at the high school. It doesn't work, but making the effort is still touching.
That utterly joyful look Connors gets on his face when he sees his new arm. It's like he's witnessing the birth of his own child there, by his reaction.
Spider-Man taking off his mask to calm down a young boy, trapped in a car suspended from a bridge by some webbing, and telling him to put the mask on because "it will make you strong" in an attempt to get the boy to climb up the car so Spidey can pull him out. And it works. It's also entirely possible that the mask was supposed to help protect the boy from inhaling the smoke produced by the flames around him, which would make it heartwarming and awesome simultaneously.
The boy's father who lines up the tower cranes of Manhattan to provide a series of swing points for the wounded Spider-Man to reach Oscorp Tower. Well, not just he alone: the boy's father mobilizes several construction teams, and, instead of evacuating the city as they've been told to, they all stay and make sure Spider-Man had as easy a time as possible getting to the fight. Average Avengers, right there.