Currently the number, scope, and names of quests have not been revealed because the game is not out yet. However, we certainly know a great deal about the main quest line as well as two side quests that have been officially publicized with minimal details as part of the marketing materials provided by Avalanche.
Main Quest
All videos provided as of January 2022 indicate that the main plot of Hogwarts Legacy revolves around the rumored existance of a "goblin rebellion" lead by a highly-talented goblin named Ranrok. There also may be a partnership of sorts between the rebellion and a band of dark wizards who operate in and around Hogsmeade, lead by Victor Rookwood. Establishing the existance of such a rebellion seems to be priority one while ultimately defeating Ranrok by thwarting his efforts would be step two.
Side Quest: Natsai Onai
Requirements: Companion, Natty
It turns out that the Gryffindor companion knows a little bit more about Rookwood and his right-hand henchman Harlow than she is letting on.
Side Quest: Poppy Sweeting
Requirements: Companion, Poppy
Ever on the lookout for poachers and their traps, Poppy ropes you into several small skirmishes that will disrupt their ability to capture and sell their prey.
Side Quest: Sabastian Sallow
Requirements: Companion, Sabastian
It has been revealed that Sabastian Sallow is one of the students with whom you can team up in the game. He belongs to Slytherin house. His sister has inhereted a "debilitating" curse and Sabastian is so desperate to cure her that he has resolved to dabble in dark magic in his search for an answer. The existance of this quest was revealed on August 23rd, 2022 as was a sample dialog option. During the course of the quest, Sabastian and the main character must bypass a magically locked door. To do so, one of them must use the curse "Crucio" on the other, and it's up to the user to decide which of them will be the caster and which will be the target. Another as-yet-unidentified member of the Hogwarts community finds them in the cave and they are beseiged by inferi.
Side Quest: Haunted Hogsmeade
Requirements: Playstation 4 or Playstation 5 (until Febraury 10th, 2024)
As part of a special Playstation State of Play event that was broadcast on September 12th, 2022, it was announced that there will be a timed-exclusive side quest available to Playstation players (PS4 and PS5) for exactly one year after launch. The impetus behind the quest is unclear, but during the quest we meet Casandra Mason, a local landlord. The main character inquires as to the status of her shop and is informed that Casandra has had a "string of bad luck" with recent tenants.
Upon further investigation, the main character descends into a tunnel system beneath the building and encounters an enemy named Fastidio and his mannequin minions. Apparently we also meet a minion of Casandra's named Penny.