I am currently reading this series, and I am nearing the end of The Shadow Rising. Siuan was just stripped of her Amyrlin Seat position and stilled because she hid the fact that Rand is the Dragon Reborn, and secretly helped him. Why did her and Moiraine not come out as soon as they knew that he was The Dragon Reborn? They were only doing what was needed. I'm confused as to why they kept it such a secret from the other Aes Sedai, and then why everyone was so horrified.
TL;DR: They didn't know who they could trust within the Aes Sedai.
Siuan Sanche and Moraine Damondred were both considered "moderates" in the Aes Sedai political structure, if not outright "rebels". They had both known from a young age (their time together as apprentices, in fact) that the Dragon Reborn would be found at some point, and had been taught by their mentor that the man would need to be supported and guided if the forces of good were to win The Last Battle.
In contrast, the majority of the Aes Sedai believed that ANY man who could channel - especially the Dragon Reborn - must be gentled and/or controlled. Add to this mix the mysterious Black Ajah, who wanted to control Rand at best and kill him at worst, and you have quite a bubbling pot of risk. Announcing Rand's identity to even the senior sisters, an unknown number of whom were likely Black Ajah, would be tantamount to throwing him into the wind with no control of which way it's blowing.
In short, Siuan & Moraine were trying to protect Rand - not for his own sake, but so that the side of the Light wouldn't lose its most valuable weapon. At the time that the Black Ajah made its move and everything hit the fan, they were considering (Siuan especially) on bringing others in on the secret but weren't quite sure who to trust yet. Once their secret was exposed, it was a moot point, since those who wanted to control or destroy Rand were now in power and could spin it any way they wanted. At that point, even sisters who might have agreed with their tactic otherwise now saw it as further evidence of their treason.
I believe that their main motive for hiding their knowledge of Rand being the Dragon Reborn was that they didn't know which Aes Sedai to trust. They knew that the Black Ajah exists, and that it hunts for the Dragon Reborn and has killed other Aes Sedai that were searching for the Dragon Reborn, as well as men (and even boys) who were suspected of being the Dragon Reborn. This is told in New Spring (both the short story and the novel).
I seem to remember that just prior to being deposed Siuan was planning on telling the Hall of the Tower about Rand. I think it was just when she got the news that the Stone of Tear has fallen.
There are several reasons that have been already discussed that I won't go into, other than to say that I agree with them but there are other reasons too. For one, you must remember the whole reason the White Tower was founded was to guard against the Dark One until Lews Therin was reincarnated. Other than the Last Battle approaching, the Dragon Reborn would rebreak the world. Controversy could kill him faster than anything when dealing with Aes Sedai. Also, the Black Ajah wanted him dead since they'd been killing any boy of the right age who showed signs of being able to channel, so what would happen if he never even grew up. Also, they knew that hiding the existence of Rand was treason, so in essence they were sworn to secrecy and I'm sure the Three Oaths hold on all promises. And don't forget, a Black Ajah member almost killed Moiraine just because she suspected she knew something she shouldn't. The secret they kept wasn't just about Rand, it was for their own safety too.