On the first day of Rosh Hashana, the Minhag of Aram Soba is to sing a Pizmon upon taking the Torah out of the Aron Kodesh. The song, "Ozreni El Hai," is connected to the Aseret Yemeh Teshuva (Ten Days of Repentance), as Rosh Hashana is the first of these days. On the second day, the song "Raba Sav'ah Lah Nafshi", is sung. There are also other Pizmonim, and one should follow the custom.
One should make an effort to view the letters of the Torah from close enough to read the words. This draws a great spiritual light onto the person. Ideally, he should focus on a word that begins with the same letter as his name does. One should bow before the Sefer Torah. Some bow the amount of times as there are Aliyot to the Torah. Therefore, on Yom Tob, they bow five times, on Yom Kippur six times and so on.
On the first day, the Torah portion is the passage dealing with the birth of Yishak, because he was conceived on Rosh Hashana. When the Oleh approaches the Torah, he should look at the place from where the reading will begin. Some have the custom to take their Sisit and kiss that spot. This is considered endearment of the Misva. Even if the Oleh is not reading out loud from the Torah, he is required to read quietly with the Ba'al Koreh, in order that his Beracha not be in vein. He is also prohibited from interrupting with talk until after he recites the final Beracha. The custom is not to close the Torah between Olim, just to cover it.
On Shabbat it is permissible to add Olim, beyond the standard seven, if there is a good reason to do so. On Yom Tob, the Rambam and Maran rule that one may also add Olim. The custom of the Bet El Mekubalim was never to add Olim, in order to preserve the mystical significance of the original number of Olim. The Ba'al Tokea is customarily called to the Torah. The custom is to sit during the Torah reading, in accordance with the teachings of the Rabbenu HaAri.
The duration of Torah reading, as long as the Torah is open, is considered an Et Rahamim (time of mercy), in which prayers are answered. One may offer his prayers in between the Olim. This is why the Olim customarily pledge large sums to Sedaka at that time, and the MiSheberach for the sick and the Ashkava for the deceased is recited. The Oleh should also make a personal request on his own behalf. It goes without saying that talking during Torah reading is strictly prohibited. Moreover, doing so squanders a tremendous opportunity for Yeshuot.