If Rosh Chodesh is a "Zeman Kapara" (time of atonement) as we say in the Mussaf prayers, why does it contain no selichot and no tachanunim? Why doesn't it (the day of Rosh Chodesh itself) resemble Yom Kippur in that way, which is full of selichot?
The kapparah of Rosh Chodesh is different than the regular type.
On one level, the goat offered as a chatas on this day (also on the other Yamim Tovim, incidentally, except for Yom Kippur) is to atone for cases of tum'ah involving the Beis Hamikdash or sacred foods where there was "no knowledge at the beginning or the end" - in other words, the person never knew that he was tamei before he entered the Beis Hamikdash or ate the food, and never found out afterwards either. (Shevuos 2a)
So perhaps this makes it inappropriate for us to substitute for this tachanun or selichos, in which we regret the sins that we do know of (which we may have done deliberately or inadvertently) and ask Hashem to forgive us for them.
On a deeper level, too, there is the idea that the kapparah is not for us but, so to speak, for Hashem's having diminished the moon's size* (Chullin 60b). Again, then, this wouldn't fit with us mentioning our own misdeeds.
(What about the phrase לכל תולדותם, apparently implying that the kapparah is "for all generations" of Jews? Kuzari (3:5), cited in Beis Yosef (Orach Chaim 423), states that indeed תולדותם means "the rebirths of the months" - i.e., that Rosh Chodesh is a time of atonement for the entire month, and perhaps also this includes the idea that it is an atonement for the very fact that the moon needs to reborn each month.)
* What does this mean? A lot of different explanations have been given, but here's one, from Chabad.org. Key paragraphs:
...But the moon--even while still a "great luminary" itself--saw a deficiency in the status quo.
You're right, said G-d. My entire creation is founded on the interplay between giver and taker, between the satisfied and the hungry... As you say, dear moon, two great luminaries won't do. Become small and dark and beg your light from the sun.
G-d then tries to explain to the moon that the recipient's role is just as great as the giver's...
And how does G-d feel about the whole thing? He feels terrible. After all, He set up the moon to it, as He set up all the nudniks, kvetchers and beggars of His world. It is He who imbued them with the hunger and sense of inadequacy that makes them so gloriously great while preventing them from comprehending their greatness.
Forgive Me, says G-d. One day you'll see that it was all worth it, but until that day comes, I acknowledge My need for atonement.