It is clear from the Gemara in Succah (37b) that although the main mitzvah of lulav is the act of taking it, and once one picks it up he fulfills his obligation, nonetheless, the fulfillment of the mitzvah in its entirety with all the hiddurim is only through doing the nanuim [shaking] as well.
The Gemara brings two reasons for doing the nanuim: 1) מוליך ומביא למי שהארבע רוחות שלו. He moves them to and fro to dedicate them to He Whom the four directions are His. He raises and lowers them to He Whom the heavens and earth are His, 2) מוליך ומביא - כדי לעצור רוחות רעות, מעלה ומוריד - כדי לעצור טללים רעים. He moves them to and fro in order to request a halt to harmful winds, storms and tempests that come from all directions; he raises and lowers them in order to halt harmful dews and rains that come from above.
Below we will discuss the order of how to shake the lulav, based on the Shulchan Aruch (651: 2 - 9), the Mishnah Berurah, and various other sources.
The Correct Way to Hold the Arbah Minim Whilst Reciting the Berachah
One must hold the arbah minim the way they grew on the tree, and if one does it the opposite way, he doesn't fulfill his duty. One takes the lulav, which has three mitzvos in his right hand, and the esrog which is one mitzvah in the left. If one took all four minim in one hand, there is a machlokes haposkim if one fulfills his duty. Due to doubt, one should take them again without a berachah.
For one who is left-handed, according to the Shulchan Aruch he should take the arbah minim the same way as everyone else, and according to the Rema he should take the lulav in his right hand (i.e. the left hand). These dinim apply both during the berachah, and during Hoshanah's.
Making a Berachah Before the Mitzvah (עובר לעשייתן)
One should recite the berachah על נטילת לולב and שהחיינו, before the mitzvah. However, one should be careful not to make it עובר דעובר לעשייתן, too far in advance. In the poskim we find three ways to carry out the aforementioned halachah:
1) To hold only the lulav in one's hand, and recite the berachah, and only after the berachah does one lift the esrog. This is the opinion of the Bikuray Yaakov (s.k. 22). This was the practice of Rabbi Shlomah Zalman Auerbach, zt”l (Halichos Shlomah, Perek 11, sif 18), he would leave the esrog on the table in front of him until after the berachah.
2) Hold the esrog upside down with the uketz facing upwards, make a berachah, then turn the esrog around and have the uketz facing down and the pitom upwards. This was the minhag of the Steipler (Orchos Rabbeinu, Vol. 2, pg. 289, ois 3)
3) Have positive kavonah [intention] not to fulfill the mitzvah until after the berachah. The Gaon writes that this is the best solution. This was the minhag of the Chazon Ish (Orchos Rabbeinu, Vol. 2, pg. 289, ois 3). The Chayei Adam (Klal 148, sif 10) writes that it is better to have kavonah not to be fulfil the mitzvah, than to turn the esrog upside down, as turning it upside down can cause damage to the esrog.