There is a Yiddish song, “Sheyn Vi Di Levone,” “Nice Like the Moon,” and your correspondent has been singing it (the Lainie Kazan version) since learning about Kiddush Levanah. One is a love song, and the other is an ancient practice.
In terms of the religious practice, we recite Kiddush Levanah every month as the moon begins to wax. Of course, we celebrate Rosh Hodesh when we cannot see the moon at all, but four days later, when we would be able to see it, after the evening service the minyan (optimally a minyan) would go outside and look for the moon. If we cannot see it, the prayer is not recited. If we can see only a part, or if the weather is bad and the minyan is inside, we may still say it. If we have not seen it every day and we have gotten past the 14th day of the month, we give up on saying it until the next month, as the moon will be waning at that point. In Tishrei, we wait until after Yom Kippur. In Av, we wait until one day past Tish’ah BeAv.
The rabbis mentioned the ritual in the Shulhan Arukh (Orah Hayyim 426). So what do we do to celebrate?
We recite Psalms 148:1-6. We recite birkat halevanah (no surprise). The blessing (we paraphrase here) praises God for creating the skies and all Heaven’s host, and appointing roles and times which they follow like clockwork at God’s bidding, and we are glad God has bid the moon to renew itself. We continue with singing “David Melekh Yisrael” and then we greet at least three persons with “Shalom Aleikhem” “Aleikhem Shalom.” We sing a few rounds of “Siman tov umazel tov,” and then recite Aleinu. If there is a minyan present, we say Mourner’s Kaddish. Additional singing and dancing may ensue.
Debbie Friedman wrote a song for this in 1998, available at this link. And yes, of course there is a BirkatHalevanah.com — one may find blessings from the Sephardic Heritage Foundation there. We get to praise God for Creation actually standing outside and appreciating it - it doesn’t get much better than that!