The young Beethoven, then just under 22 years old, left Bonn on November 2nd or 3rd, 1792, and arrived in Vienna around November 10th. After finding lodgings and acquiring a piano, he began his studies with Joseph Haydn in December 1792, as documented by a notated bill for "Haidn 8 groschen." These 8 groschen were equivalent to about 24 cents, a symbolic amount Haydn typically charged non-aristocratic students.
Haydn had already met Beethoven five months earlier, when he passed through Bonn in July 1792 on his return journey after a successful 18-month stay in London. Impressed by Beethoven's talent, arrangements were quickly made for the young composer to travel to Vienna at the city of Bonn's expense. The original plan was for Beethoven to study with Haydn for a year or two, make a name for himself in Vienna, and then return to Bonn to serve as a musician at the Elector's court, following in the footsteps of his father and grandfather. In fact, Beethoven would never return to Bonn.
Vienna was then the center of the classical music world, a vibrant hub that fostered musical innovation. Here, Beethoven hoped to hone his skills under Haydn's tutelage. Despite his busy schedule and advanced age, Haydn accepted Beethoven as a student, recognizing the potential for greatness in his young protégé. However, Beethoven quickly became dissatisfied with the lessons, and they proved unsatisfactory for both student and teacher.