I know that Project Tsuki-no-Me was supposed to be "Madara's" Sharingan's reflection on the Moon itself, as Tobi declared during the Five Kage Summit. However, when he got his hands on the Nine-Tails Chakra from Lords Kinkaku and Ginkaku, the moon turned into a red one — it looks just like a regular full moon, but in red colour. So, is that the first stage of Project Tsuki-no-me?
The Sharingan has the ability to hypnotize someone just by making eye-contact with them.
From the Sharingan article:
The Sharingan's final commonly-used ability is capable of inducing a unique brand of hypnosis that involves suggesting actions and thoughts to the opponent through genjutsu cast by simple eye contact between both parties
Obito was even able to control Kurama, commonly known as the Nine-Tails (九尾, Kyūbi), so you can imagine what Madara's Sharingan hypnosis is capable of.
Again, from the Sharingan article:
[...] only Madara and Obito Uchiha have achieved this feat. It has been stated by Madara Uchiha that the beasts were slaves to those with the blessed eyes. When this was used against Manda and the Nine-Tails, their eyes took on the tomoe pattern of the users' Sharingan while under hypnotic influence
In order to hypnotize everyone, Madara needs everyone to look into his eyes. Therefore, he utilizes the moon as a reflector for his eyes so that he can apply Sharingan hypnosis on everyone who looks at the moon. The red moon is the start of his Sharingan reflection.
You might also want read this answer about Infinite Tsukuyomi: Why is everyone affected by Infinite Tsukuyomi?
Tsuki no me, is essentially a permanent dream state, the user's body is kept alive by a certain mechanism- spoiler avoided. But the "spirit" is tricked into a "happy" dream.