This is a brief explanation of both the magical and non-magical mechanics you can use in Hogwarts Legacy.
Broom Flight/Mounts
Flying on a broom is an important aspect of the wizarding world. It should surprise no one that in a game with a "broom-flight professor" that users will have the option to travel by broom. While certainly slower than travel by floo flame, brooms offer the added flexibility of being able to get off at the moment and location of the user's choosing--including onto an elevated position such as a roof.
While on a broom, there is a meter in the bottom right-hand corner that displays the "charge" of the broom. Activating the boost allows users to fly into the air for a very short amount of time while the charge is used up and then return to normal floating height once the boost deactivates. Users can level this mechanic in order to remain in the air for longer.
It was confirmed during the second gameplay showcase that there is a merchant in Hogsmeade at the Quality Quidditch Supply shop who wants to improve the quality of his brooms and is willing to allow the student to test-fly his inventions. One would assume that whacky hijinks abound.
Climb
While we have not yet seen the climbing mechanic, this ability was confirmed during the Question and Answer portion of the AutoDesk event on August 3rd, 2022.
Feed and Groom
One of the most important parts of caring for fantastic beasts is getting them to trust you, as they then will allow you access to magical materials you can weave into your clothing to become super-awesome. To gain their trust, you must kidnap them back to your Room of Requirement, feed them, brush them, love them, give them toys, and then they will provide you with the goods. Much of this can be automated if you, like me, are a psychopath with a ton of moonstone.
Jump
This ability was confirmed during the Question and Answer portion of the AutoDesk event on August 3rd, 2022 and was first seen via tweet on December 7th, 2022.
Run/Walk
During the gameplay reveal trailer, the main character's avatar can be seen both moving at both run and walk pace.
Spellcasting
There seem to be several ways to learn spells: via class, via books, via talking with "friends" who are eager to torture you. After you learn a spell, you can add it to your spell diamond and then cast it at appropriate times. The spells themselves are covered under "spells" in this guide. However, it's important to remember that some of the spells are contextual, meaning that they are NOT available at-will, but rather at specific times as is appropriate for the story.
Swimming
While you cannot dive, it was revealed during footage obtained from the Hogwarts Legacy preview event on January 18, 2023 that the main character can swim in the rivers and lakes that dot the Highlands. Also, there's the dirty tree water in the greenhouse and the jaccuzi slime that is the Prefect's Bathroom.