Whenever I'm outside, and I happen to know where the Moon should be, I look for it. Sometimes it's a waning gibbous phase, low in the west at sunrise. Or it might be a half Moon, high in the south at sunset. My favorite is the thin crescent, a "fingernail" Moon, hanging near the horizon.
The thinner the crescent, the harder it is to see. That's because when it's really thin it's just hard to find, of course, but also because the thinner it is, the closer it must be to the Sun in the sky. We see lunar phases because the geometry of the Earth-Moon-Sun system changes, and the phase depends on where the Moon is relative to the Sun. A thin crescent is always surrounded by bright sky, and is also near the blinding Sun. That's quite a gauntlet to pass!
It's tough to photograph, too, which is why that image above is so remarkable. The Moon was about 24 hours from being new -- meaning that in only a day it would be at the closest point to the Sun in the sky -- when that picture was taken. The picture is eerily lovely, but to my eye, is all the more cooler because of the circumstances under which the photographer got it. The story is written on the wonderful Lunar Picture of the Day website (tip o' the dew shield to Chuck Wood from LPOD -- with whom I recently had lunch -- for the picture). Give it a read-- you might want to try for a young Moon sometime yourself!