Quick recap of season 1 of DGM for anyone who needs a refresher, since it was requested often in the reddit Episode 1comments. Note my summaries do not do justice to the humor, wit, and depth of season 1. I’ve left out all side characters, characterizations, and anything not related to the overarching arc:

World setting:
The War- The Millenium Earl vs The Black Order.
Main protagonists- The Black Order has exorcists who wield Innocence and use it to defeat akuma. The Millenium Earl and his Noah Clan kill people using akuma. They believe the Noah are the true disciples and that the Exorcists follow the false God. These 13 Noah disciples plan to bring about the end of the world.

Intro to series: Allen Walker is a brown-haired boy with a dysfunctional red arm. He was adopted by Mana Walker, a traveling clown, after they met at a circus in England. After a hard life in the circus as an abused errand boy, Allen loved Mana and his new life together.  When Mana died, a figure called the Millennium Earl approached Allen and offered a contract to bring back Mana from the dead. Allen happily accepted. Little did Allen know that the Millennium Earl preyed on people’s grief to create akuma, in which the souls of the deceased were called back into skeleton frames and trapped in the body of the living greiving human. After murdering the human and crawling into their skin, the akuma were forced to obey the Earl. Mana’s soul returned, but cursed Allen for calling him back and attacked him to take over his body, Allen’s red arm instantly transformed into a claw and destroyed Mana’s soul. Mana’s curse left a pentacle scar on Allen and his hair changed into stark white from his shock of destroying Mana’s soul.

Afterwards, Cross Marian came and took in Allen as his disciple. He revealed that Allen’s arm was “Innocence”, a weapon considered the Crystal of God and it was the only way of defeating the akuma, as well as exorcising the souls trapped within them. Mana’s curse caused Allen to develop the ability to see and hear the trapped suffering souls crying within Akuma, which helps him identify and exorcise Akuma, which otherwise are disguised as normal humans. Allen grew up as a polite young man who gave off the impression of being a “proper English gentleman” and has a habit of being distant with others and using formal language at all times. Unfortunately Cross had a bad gambling, womanizing, and drinking habit. He would leave all his debts in Allen’s name resulting in Allen having to earn money in shady ways during his time with Cross. Cross trains Allen harshly, giving him humorously impossible tasks that leave Allen in fear of Cross and traumatized. One day, Cross told Allen that he was ready to become a true exorcist and to go seek The Order. Cross refused to go back with him to headquarters, and instead knocks out Allen with a hammer and disappeared, leaving behind his yellow golem Timcampy with Allen to lead the way.

After traveling to the Order alone with Tim and misadventures along the way, Allen arrives and first meets Kanda Yu, Lenalee, and Komui Lee. Due to a misunderstanding regarding Allen’s arm, Kanda attacks Allen believing him to be an akuma intruder. They have bad chemistry from the first meeting and continue to bicker throughout the series. Allen gets along with everyone in his new home, befriends the chef Jerry, and members of the Science department such as Johnny, the guy with the glasses. He's sent on many mission as he gets closer to other members (ex. 1st Mission with Kanda, 2nd Mission with Lenalee) and meets new members (Krory, Miranda, Chaoji). He vows to live by Mana’s saying “Keep on walking” and swears to save the Akuma souls by exorcising them…"

Season 1 explores several problems with the premise.
First the miraculous "Innocence" is typed as either a parasitic or equipment type. The parasitic type drains the wielder in exchange for greater power compared to the equipment type, and overall shortens their lifespan. If the Innocence feels betrayed by the wielder and their synchronization rate (affinity to their Innocence weapon) falls below 0%, the Exorcists becomes a Fallen One, which is fatal and essentially morphs them into a monster that expels massive amounts of energy that kills the wielder and anyone nearby. After the Exorcist's body is used up, the Innocence reforms is neatly available again to be potentially wielded.

Second, the Noah Clan, although they are sadistic and messed up, are shown to be likable characters. Road Kamelot, who controls dream dimensions, in particular grows to like Allen, and despite a rough history (she crushed his eyeball with a spike, tortured him and his friends repeatedly), she's made it clearly apparent that she has an interest in him.

Third, Allen in the beginning is introduced as an "proper English gentleman". He's always polite, to the point of being distant. He has a major savior complex and self-sacrificing to a fault. Kanda, Lenalee, and Lavi in particular notice this trait. Allen's goals are to save the akuma souls while not harming any humans, including the Noah. In contrast, most of the Order hate Akuma, since most of them joined the Order after suffering the loss of their family members due to akuma (for example Lenalee and Komui's parents were murdered by akuma). For these reasons, Allen's attachment to the akuma is viewed with disgust and suspicion from characters like Chaoji.

Summary of events right before Hallow
Finally, near the latter half of the 1st season, Allen and Co. are tracking down his master Cross Marian due to recent assassinations of General rank Exorcists. Cross Marian has discovered that on an artifact called Noah's Ark, the Millennium Earl keeps a mysterious objected called the Akuma Egg. The Ark itself appears as a small city and doors inside connects different places in the world with a gate. Allen and Co. enter the Ark. A mysterious figure called the 14th disciple is revealed to have stolen control of the Ark from the Millennium Earl years ago. As the Millennium Earl orders Road to construct a new Ark, destroy the old one, and download the new Egg. Allen, Kanda, Lavi, Lenalee, Chaoji, and Cross find themselves trapped in the Ark as its being destroyed. Cross then commands Allen to use the Powers of the Musician to stop the download, without explaining the powers. Cross connects Allen to a secret room in the Ark, with a piano. His constant companion golem Timcampy projects an image of a musical score written in code. He remembers the code as something made up and shared by his dead adopted father Mana. Allen realizes that he has the song within him and a strange hallucination of a shadow instructs him to use the Ark. Allen successful stops the download and saves himself and his friends. Allen gains the power to use the Ark and the Order’s Science Division takes the Akuma egg to investigate it’s properties.

Despite his misgivings that his Exorcist Master Cross knowing the path to the Musician’s room in the Noah’s Ark, Timcampy holding a score written in a play language made-up by Mana, and Allen’s own subconscious ability to easily wield the Ark, Allen doesn’t have a chance to ask Cross Marian despite attempts to track him down.

The Order doesn’t trust Allen’s abilities and assigns Inspector Howard Link to shadow him and watch his movements. The Noah attack the Order’s headquarters to take back the Akuma egg. The Headquarters is destroyed during the attack and it becomes easily apparent that the Millennium Earl could have destroyed the Order at any time. Before the Akuma Egg is taken away, Allen successes in destroying it, but not before Howard Link is able to retrieve a shard.

Hallow opens with Allen and Co. moving to the new Headquarters and Allen finally getting time to talk to Cross.