The first quarter moon is halfway between new and full moon. Half of the moon is illuminated and it's a quarter of the way through it's orbit around the Earth.
A waxing gibbous moon is in the sky before it's dark and lights up the early evening. It's more than half lighted, but not quite full. It sets after midnight.
The new moon generally can't be seen, except during a solar eclipse. The moon crosses the sky with the sun during the day, and its shadow side faces Earth.
January's new moon will be the biggest eclipse miss of 2024, an ultra-thin crescent not nearly aligned closely enough with the sun for a solar eclipse.