Volume 1 covers roughly the events in the first OAV, so I'll try to be brief when talking about things that are already in the OAV. However, I don't always know when to stop writing, so this is likely to be longer than it should be...
"I Want To Go Back..."
The first portion of this volume is pretty close to the OAVs: Alice is on her way home from shopping when she is accosted by Rin. The next morning he tries to make it up to her by giving her some gum for real, but suspecting a trick, Alice knocks it out of his hand without looking. When she realizes it really is gum, she apologizes, but Rin begins to bawl his head off. She takes him to a shop to buy him something, and when she asks what he wants, he yells "GUM!"
At school, Alice still can't remember her classmate's name, and she is reminded that it's Suzuki Mariko. Mariko suddenly notices gum in Alice's hair and points it out. This attracts the attention of the other girls, whereupon Alice is rushed off to get her hair cleaned. This in turn attracts the attention of Ogura Jinpachi, who thinks it's rare for Alice (still considered a transfer student) to be the center of attention. After helping Alice with her hair, the girls tells her they had wanted to talk to her long before that, but had been reluctant to because of the "mature mood" that Alice seemed to possess. They also tell her that Jinpachi has been watching her for some time now. Moving on to other topics, they ask her about her unusual name, and she tells them that her mother was a big fan of the book Alice in Wonderland. One of the girls jokingly asks if Alice can also talk to animals, like the fictional Alice, and she replies that, no, but she can talk to the trees and flowers. There is an awkward pause, but the girls get over it.
After school, Alice waves goodbye to the other girls and starts home in much better spirits. She suddenly realizes what time it is, and that she'll be ambushed by Rin again, so she turns back to stay at school a little longer. True to her nature, she wanders to the garden in back, and it is there that she overhears the embarrassing conversation between Jinpachi and Issei.
The next day Alice has to babysit Rin. In the ensuing struggle, he blurts out that Alice isn't a virgin. Embarrassed and offended, she stops, while Rin goes on to explain that he knows what the term means. Of course, he promptly proves this to be untrue when he explains that his mother is a wonderful virgin, but he knows that Alice isn't. Later, after the incident with the cat, Alice takes Rin to the zoo, where he embarrasses her again by yelling the word "virgin" in public. Of course, things get worse when she bumps into Jinpachi and Issei. At the coffee shop, Alice slips up and asks if the two are there on a date. Red-faced and sputtering, Jinpachi explains his brother works at the zoo, then follows with a lengthier explanation about his and Issei's dreams. This is slightly different than the OAV: in the manga, they only talk about themselves as Gyokulan and Enju; there is no mention of the others as yet.
Later, Rin's jealously comes to a head, causing the accident where he falls off the balcony. At the hospital, Mrs. Kobayashi slaps Alice, then everyone breaks down crying. At the Sakaguchi's, Mr. Sakaguchi comes home and asks after both Alice and Rin. His wife tells him Alice is already sleeping, and adds in wonder that miraculously, Rin suffered no serious injuries because the bushes and trees beneath him had cushioned his fall. That night, Alice's sleep is disturbed by nightmares of seeing Rin fall from the balcony.
Alice wakes up feeling ill with the thought of what she did, but Hajime comes out and comforts her. He relates what their mother said about Rin, and adds that even here, the plants are on Alice's side. She goes to school, where she is treated coldly now by the same girls who were her friends the day before. Finally seeing his chance, Jinpachi steps in to help.
"Awakenings and Dreams"
In brief: Alice leaves cookies and flowers in front of Rin's hospital room; she prays to God to help Rin recover somehow; a voice says "From here on, we will watch over and protect the Earth." Rin's eyes suddenly open and a tear trails down his face.
At school, Alice is eating lunch outside when Issei spots her. He tries, without too much success, to engage her in conversation. Hearing his name called out, Issei says, "Oh, he's here." Alice suddenly realizes he's waiting for Jinpachi and hurriedly starts to leave. Issei is trying to pull her back when Jinpachi barrels around the bushes and runs right into his arms, yelling "I'm in shock!" He then notices Alice. Issei just says, "You moron." It is at this point that the two of them talk to Alice some more about their dreams, and go on to list the rest of the seven people on the moon. They mention that the seven scientists' language bears an uncanny resemblance to Japanese in some ways, and that's why all seven had names that sound like Japanese flower names. Issei scratches cartoon representations of the seven people, accompanied by constant complaining from Jinpachi about how bad his drawings are. Jinpachi finally mentions what he is so upset about. In his dreams last night, Mokulen (whom he loved) got married to Shion. Once again, he throws his arms around Issei. After he calms down, Alice asks what they plan to do, to which Jinpachi replies that they would like to find the other five people. Issei jokes that they can form a dream club at school.
Lunch break is over, and they head back to class. But in class, Alice can't get the other two's story out of her head. She keeps thinking about Gyokulan and Enju, but since she hasn't seen them, she keeps attaching Jinpachi's and Issei's faces to the image. When the two finally kiss, she can't help but be reminded of AIDS. It is in this state that she comes to realize her teacher is calling her. He tells her there is a telephone call for her. The events that follow are essentially those in the OAV: she goes to the hospital and finds that Rin is awake.
Mrs. Sakaguchi tells Hajime to call Alice for dinner later. When he doesn't find her in her room, he looks outside and thinks, "I thought that might be it." What he sees is: with no wind blowing, the trees are swaying gently, and Alice is talking to them. He thinks back to a long time ago, the first time he saw Alice talking to the plants, and remembers that he used to envy her for that.
At school, the girls are discussing their previous treatment of Alice and feeling fairly bad about it. Each girl, perhaps afraid, is somewhat reluctant to apologize until Jinpachi walks up and suggests they do it together. Just before class starts, when Alice walks in, they stand together and shout out at Alice, "Sakaguchi-san! We're sorry! Good morning!" Jinpachi has to suppress his laughter.
That afternoon, Mrs. Kobayashi visits to talk with Alice's mother. She is clearly ill-at-ease, and has trouble explaining the problem. Finally, gathering up her nerves, she explains that Rin told her that, no matter what, he wants to marry Alice. Mrs. Sakaguchi sits in shock. Meanwhile, Alice is walking home from school and feeling a lot better. She is no longer afraid of Rin, and talks to him when he approaches. This is pretty much the "kissing scene" in the OAV. This leads to her mother questioning her about the marriage to Rin, Alice's fainting, and her dream of Mokulen and Shion.
In the Kobayashi household, things aren't much better. Rin's mother tries to talk him out of his idea of marrying Alice, or at least to wait until he's older, but to no avail. He starts off, "And in the meantime, what if Alice and some strange guy..." and breaks down and cries. His mother, distraught, consoles him and tells him she understands. Later she tells her husband that Rin has been acting more and more spoiled lately.
The rest of the story is closely detailed in the OAV, with two minor notes: When Rin jumps off the balcony and lands, he thinks to himself, "10 points! "; as Takashi is speeding around the bend in the road, there is a thought bubble above his head of a bowl of ramen (one reason I can't watch the video without laughing at this point).