"A Wild Hare" (1940) is considered the first to feature both Bugs' and Elmer's catchphrases - "What's Up, Doc?" and "Be vewy quiet...I'm hunting wabbits" respectively. Actually, this is a great cartoon, but "Porky's Hare Hunt" (1938) is the first BB cartoon...followed by "Hare-um Scare-um" (1939), "Elmer's Candid Camera" (1940) and then "A Wild Hare (1940)."
This cartoon is, indeed, the first time he was called "Bugs Bunny." It was also the first time he was less of a "hippity-hoppity" rabbit than he was in his 3 earlier movies and closer to the rabbit he is today.
Got carrot?