To get a get a broom in Hogwarts Legacy you'll need to complete two missions: Flying Class and Flight Test. Once you've completed those you can start flying all over the map.
Unsurprisingly, flying is an great way to get around given the size and scale of Hogwarts Legacy. But before you can take to the air you'll have to complete certain parts of the main story and master specific Hogwarts Legacy spells. Only then can you take your pick of the brooms at the Sprintwitches shop that services all your 'stick needs.