Hey, Amagami was originally a Dating Sim, so of course it's a given that it has heartwarming moments.
Every single scene before and immediately preceding the rolling of the end credits. Naturally, the 'Bad Endings' are the exception, but for only some of the girls. Here's the list of girls and their following endings from best to worst (WARNING - spoilers abound):
Morishima Haruka - Popular girl who rejects every boy who confesses to her. Part of her story (and her appeal) will be getting to know her despite being rejected and being her cute underclassman (she likes cute things).
Best End: Morishima reserves a hotel room on the pretext that Tachibana will be meeting her parents. Her parents aren't there. She takes a shower and comes out in a towel. She's flustered about Tachibana not peeping or why he hasn't confessed again despite having confessed twice (and getting shot down both times). She tells him that she's so in love with him that she can't think straight anymore. Tachibana smiles and tells her that the clumsy upperclassman he got to know is the same girl he fell in love with. Morishima tells him to call her Haruka and they share a passionate kiss, while her towel falls off. A few years later, Tachibana becomes a police detective and Morishima is his loving wife.
Good End: Morishima occasionally spends the night in Tachibana's room while she is in college.
Normal End: Tachibana buys her a puppy. Morishima sees Tachibana less of a cute kouhai and more of a reliable man.
SS Plus Tachibana proposes in front the whole school, she accepts and kiss him.
Tanamachi Kaoru - Known Tachibana since middle school. Messes with Tachibana a lot and he considers her a "bad friend." Extremely passionate and not afraid to tell her opinion on things. In her route, she kisses Tachibana a lot and he wonders if she's just messing with his head. She's not, she has it bad for him. She may playfully bite him on the ear, thus one of at least three girls who lend the game and franchise its name "Amagami."
Best End: Her mother is getting married to another man and this is a big issue for Kaoru. Tachibana becomes her emotional support throughout. On Christmas, they spend the night in Tachibana's room. While in bed, Kaoru climbs on Tachibana, shares a kiss, and tells him how their relationship parallels that of her parents. The morning after, they can be seen riding a bike together and then meeting with Kaoru's family some time afterwards. They'll be "bad friends" forever.
Good End: They're both in college, sharing a place together. She cuts his hair. She realizes they'll be "bad friends" for a long time to come.
Normal End: Keiko (Kaoru's friend) is dumped again. Tachibana encourages her. Keiko asks him to be her boyfriend. Kaoru punches him in the face and yells that there's no other girl for him but her.
Bad End: She's crushed but forgives him by throwing him in a river during winter. Tachibana floats away while she yells that she still loves him.
Nakata Sae - Shy underclassman with big breasts. Came from an all girls school, afraid of guys.
Best End/ SS Plus: Sae feels bad about making Tachibana her boyfriend. Tachibana runs after her, telling her how much she means to him. They spend their Christmas watching a movie in a private room. They share a deep, passionate kiss. Many years later, she walks out of a hospital towards Tachibana... pregnant with his child.
Good End: Sae works in a Maid Cafe. She never would have had the confidence to work there hadn't it been for Tachibana's support. One day, she'll make Tachibana fall in love with her.
Normal End: They fool around in the school infirmary. They share a deep kiss. She tells Tachibana to lock the door. Is sex implied? You be the judge.
Nanasaki Ai - Cool girl who places swimming above everything. Sees Tachibana as a bit childish for his age... but he reminds her of her own little brother.
Best End: Her relationship with Tachibana strains her swimming performance and puts her at risk of not being qualified for the tournament. They work things out and Nanasaki manages to stay in the swim team. After the festival, they spend their Christmas in a hot spring in the mountains. Tachibana realizes how much she means to him and confesses. She accepts and hugs him while their both in the spring. Many years later, they watch fireworks from their room.
Good End: They spend Christmas at the school festival. Tachibana confesses on the rooftop. She accepts. They watch the lighting of the Christmas tree together.
Normal End: Many years later, they're both working in different companies. Nanasaki is late meeting Tachibana at a noodle stand because her boss chewed her out for not being married at her age. Tachibana mumbles something about marriage. Nanasaki smiles and asks him what he just said again.
Bad End: Nanasaki is hurt and doesn't talk to him for a while... but eventually forgives him anyway. Many years later, they're talking to each other on a beach with his head on her lap. She realizes that she can never stop loving Tachibana.
Sakurai Rihoko - Childhood friend who eats a little too much. Has been with Tachibana since elementary school. She has had a crush on him ever since.
Best End: Tachibana spends Christmas with her and his sister in their house. Later outside, he tells her that he always took their relationship for granted, but now it's different. They share a deep, passionate kiss. Many years later in college, Tachibana buys her a creampuff (her favorite food). Inside it, there's a ring (part of her story, a made up myth about proposing with food and being happy forever) and she accepts. They've always been together and always will be.
Good End: Tachibana spends the school Christmas festival with Rihoko. He hugs and realizes how much she means to him. The next year, he joins the tea club but it's closed due to lack of members. Despite all this, she's with Tachibana and that's all that matters.
Normal End: She wakes Tachibana up while in bed by jumping in his bed and going underneath the blanket. The silly stuff reminds her of their childhood and the time they spent together. That will not change.
Bad End: Tachibana begs forgiveness from Rihoko for standing her up on the Christmas date. Rihoko is sick with a cold, but tells him she was never angry at Tachibana and never can be. They repeatedly kiss (Rihoko tells him he might catch the cold). A little while later, Tachibana begs Rihoko to punish him. She makes him eat an all you can eat ice cream parfait. He faints and Rihoko smiles. They'll be best friends forever.
Ayatsuji Tsukasa - Hard working class representative. Image is extremely important to her and acts kindly towards others. In reality, she's very distrustful of people and very cynical. She acts like a Tsundere towards Tachibana, but her love is sincere.
Best End: Ayatsuji is accused of flirting with Tachibana while planning the Christmas school festival. She reveals her real, cynical personality. Everyone begins to hate her, confirming her cynical belief that people liked her just because she was a class representative. Tachibana becomes her sole pillar of emotional support, and later Kaoru joins in defending her. She quits being a class rep and forgets about her perfect image. On the night of the school Christmas festival, she and Tachibana sneak into the school and celebrate their own party there. They spend the entire night there. She becomes truly happy.
Good End: Her true personality is never revealed and she never quits being a class representative. She plans out the party. She hugs Tachibana under the Christmas tree in front of everyone. There's a myth that whoever hugs under the tree becomes happy forever. Many years later, Ayatsuji and Tachibana visit the school. Ayatsuji tells Tachibana she's afraid... she's been happy for so many years and hopes that happiness will never end. Their child runs up to them.
Normal End: They go to the same college and share the same room. She kicks him out of bed while making breakfast in typical Tsundere style. She orders to him to always make her happy.
Part of what makes the above so effective is the beautiful BGM that is used for the scenes. Appropriately, the OST names it 'Confession'.
Risa's arc in the anime, which runs a bit differently from the game, reaches its heartwarming moment when she reveals to Junichi that she was being selfish and fending all the other girls off so she could have him for herself, and now felt guilty about it and is willing to let go of him. He would have none of it though, going as far as willing to apologize to the other girls with her.
The highly anticipated Miya episode has its own as well: after finding the lost kitten, she and Junichi try finding its mother. They almost give up when the kitten recognizes its mother and runs to it. Miya is distressed at first when the kitten bites its mother, but Junichi assures her it's the "playful bite" animals do to each other as a sign of affection and bonding between family. Miya ponders on this, and on the way home she does the same "playful biting" to Junichi, implying she too would like to bond with him like the cats did.