because I know I'm not the only one with this misconception:
new moon does not mean "oh, the moon is at night, she just doesn't appear/appears as a shadow"
this geometrically makes no sense. if the moon appears at night, then that means the sun's light must also be hitting it to some extent. so the more nightmoon you have, the more she's lit up. (Unless you have an eclipse.)
a full moon is completely in night, and sunset is directly correspondent with moonrise. a waxing moon appears behind the sun, as the moon's orbit takes her from west to east. (this is also why tidal cycles take about 24 hours and 50 minutes.) You can sometimes see a waxing crescent moon behind the sunset. In the same way, you can see a waning gibbous moon in the east during sunset, and see a waning crescent moon before sunrise.
if you want to enjoy the daymoon, quarter moon and gibbous moon are good options. The crescent moon often follows or leads too closely to the sun to be comfortable to look at. I'm also not sure exactly when she becomes really visible in the day, due to that proximity.
the easy, intuitive moon knowledge you never knew if you're like me and felt alienated from your world from most of your life. It adds a lot of personality to the moon when you understand it, I feel, and deepens your connection to her. Anyway I'm gonna go werewolf out now.