The First Viennese School is a name mostly used to refer to three composers of the Classical period in Western art music in late-18th-century to early-19th-century Vienna: Joseph Haydn, Wolfgang Amadeus Mozart and Ludwig van Beethoven.
[1] The designation "first" is added today to avoid confusion with the Second Viennese School.
These composers sometimes encountered each other: Haydn and Mozart were even occasional chamber-music partners.
Nor is there any evidence (other than Haydn teaching Beethoven) that one composer was "schooled" by another, in the way that Berg and Webern were taught by Schoenberg.
Attempts to extend the First Viennese School to include such later figures as Anton Bruckner, Johannes Brahms, and Gustav Mahler are merely journalistic, and never encountered in academic musicology.