The sefarim write that the yamim noraim begin on the first night of selichos. Rebbe Uri of Strelisk zt’l said that the וחתימה כתיבה of Rosh Hashanah begins on the first night of selichos.
On Motzei Shabbos, we say the selichos מנוחה במוצאי, and each line begins with a letter of the alef beis, in alphabetical order. The exception is ך"וסמ ן"נו which are written in the same phrase, two words, next to each other, שגבם נא, "Please elevate them..." and the roshei teivos spell ס"נ, miracle. Rebbe Pinchas of Koritz zt'l explains that we don't want to separate these letters because we want to imply that the days of selichos are mesugal for miracles.
The final line of this piyut states ניסך תראם גדולות עושה, "Show them Your miracles..." During these days of selichos, we ask Hashem to show us His miracles and to grant us all our heart's desires.
We also say on this first night of Selichos, בלילות בעומדם עתירתם רצה, "Desire their prayers when they stand at night." The Munkatcher Rebbe (Shaar Yisachar Tishrei, Selichos, 43) explains that this means they stand up to the tests (which are called לילה, nighttime). They remain steadfast in their emunah, even when encountering difficult tests. We request that in this merit, Hashem should answer their tefillos.
It is the custom of Belzer chassidim to say the entire Tehillim on this day, in beis medresh, עם ברוב (many people together). Rebbe Yissachar Dov of Belz zy'a said that his avodas Hashem on the first day of selichos is greater than his Rosh Hashanah and Yom Kippur service.
