It was a "secret" mission.
The whole mission was meant to be a 'secret' for Saruman, Sauron and anyone else whom they could not trust.
They didn't want Sauron learning of their mission.
'Had I a host of Elves in armour of the Elder Days, it would avail little, save to arouse the power of Mordor.'
From the last line, Elrond clearly states that he didn't want Sauron learning of their journey.
'If that is so, it will be some time before they can begin the hunt again. Of course the Enemy has other servants, but they will have to journey all the way to the borders of Rivendell before they can pick up our trail. And if we are careful that will be hard to find.'
The Company took little gear of war, for their hope was in secrecy not in battle.
They didn't want Saruman learning of their mission either.
'Things have changed since you came north, Boromir,' answered Gandalf. 'Did you not hear what I told you of Saruman? With him I may have business of my own ere all is over. But the Ring must not come near Isengard, if that can by any means be prevented. The Gap of Rohan is closed to us while we go with the Bearer.'
Saruman, undoubtedly, would have attacked the Fellowship sooner if he had known. After his crebain came back to tell him of their journey, he immediately mustered his Wargs to assail the Fellowship.
They didn't willingly tell Denethor of the mission.
'Do not tell him more than you need, and leave quiet the matter of Frodo's errand. I will deal with that in due time.'
Denethor only came to knew of the mission through Faramir.
They accidentally told Faramir of the mission.
'... and it's my opinion that in Lorien he first saw clearly what I guessed sooner: what he wanted. From the moment he first saw it he wanted the Enemy's Ring!'
Faramir's shrewd guessing also helped.
Overall, when secrecy failed and their mission was revealed to both Sauron and Saruman, their primary concern was to protect Frodo and the Ring. What about Galadriel?
Galadriel knew of it through the messages sent by Elrond.
'Your quest is known to us,' said Galadriel, looking at Frodo. 'But we will not here speak of it more openly.