I think the early Bugs cartoons are the best and showed his personality at its richest and most entertaining.
Elmer's Candid Camera by Chuck Jones is the first cartoon where Bugs really seems like Bugs. He doesn't quite have the voice yet, but he is basically underplayed. This is almost the exact same cartoon as Tex Avery's A Wild Hare, except that Elmer has a camera instead of a gun.
To see more images from Elmer's Candid Camera, go visit Marc Deckter's "Cool, Calm and Confident" post at the legendary DUCK WALK.
A Wild Hare - Tex Avery 1940
The same situation as Porky's Rabbit Hunt, but now Bugs is less manic than when he was first born.
He has an early version of the familiar voice, he says "What's Up Doc" for the first time but he still doesn't have his name.
There are many routines that became staples of his act. Tons of variations were done of them in later cartoons-to the point where Bob Clampett told me that Bugs became a formula, and he started to feel trapped.
Bugs fakes dying. This is a new kind of humor for cartoons. It's funny because it's so serious. It's animated by Bob McKimson. Clampett later lampooned all the situations from the classic Bugs Bunny shorts that he helped gag for Tex.
Bugs is a fag. Is everybody mad now???