First Mishna of chapter 3 of Rosh Hashan seems to be out of place since it speaks of the am yisrael and the whole beit din seeing the new moon and yet not having time to consecrate it, whereas the rest of the chapter deals with blowing shofar and other Rosh Hashana activities.
Answer may be this: When does all am yisrael try to see the new moon? When they need to know whether it is Rosh Hashana or not. So point of fact, the mishna is speaking about Rosh Hashana, not rosh hodesh. The activity of Rosh Hashana evening is for father and son to look for the moon so as to know if today is Rosh HaShana or not.
In the Yerushalmi Rosh Hashana chapter 1:3 is written a long piece of agada based on the verse Mi kaamcha yisrael goy ehad..." This agada give two distinctions of am yisrael, one that they are happy on yom din, rosh hashana and two that G-d lets them decide when the yom din will be.
On Rosh Hashan we have three prayers in the shmoneh esrai, malchiot, shofrot and zichronot. We crown G-d King, we ask him to remember us for good and we do it with a shofar. Mishna in Taanit (2:2) describes that when troubles occur to Israel, we fast and add 6 extra tefillot to the shmoneh esrei. Among those we add (mishna 2:3) are Zichronot and Shofrot, but not malchiot. Why not?
As parents when our child does something wrong that cannot be ignored, two things can happen: either he comes to us and admits he did something wrong, or we seek him out. Generally the first does not happen and we need to seek him out. And it is a pity because had he come to us the punishment would have been less, because he is admitting.
In the same way, there are two ways G-d can relate to us. Either he can bring on us hardships and then we turn to Him to save us (that is a taanit), or we of our own volition can come to him and ask him to judge us (rosh hashana). That is why on fast days we do not say malchiot because the hardship we are facing means G-d has sought us out and is judging us, he is showing that he is already King. We can only ask for mercy, i.e. shofrot and zichronot.
But on Rosh Hashana we are coming to him, making him king and asking us to judge us, and asking for mercy. The two agadot above complement each other. Because G-d has us determine when the judgemnt will be, when the new moon is spotted and rosh hashan declared, therefore we can be happy when we come to judgement. We can count on His mercy. BUt we must make the first step and be self-correcting.
Heard from Tzachi Lehman Sept 2011