I understand that the WASD keys are standard for moving a character along two axes (left/right, up/down) because in the QWERTY layout, they are arranged like the arrow keys. I want to make a game where the character can move along three axes, i.e. also front/back. Are there standard key bindings for this as well?
For something with rotation its generally
wasd for throttle and roll,
shift+wasd or uhjk for strafe orthogonal to forward.
For purely translation, then wasdec or wasdqe.
But its ultimately the situation that dictates what is best used.
Don't think "I should use the standard", think "Does the standard suit this?"
WASD is just a de facto standard for FPS games. If you play an FPS from the pre-Half Life era, you'll end up with all sorts of variations. That said, the analogue from 5DoF to 6DoF is jump/duck, which varies for each game. Case in point, Quake used jump and duck for up/down movement underwater, even though it used arrow keys for forward/back and turning, with alt+left/right for strafe.
The original 6DoF FPS, Descent, had flight-sim like controls and used the +/- keys for up/down.