The common practice when taking the four species on Sukkot is to first take hold of the Lulab (of course, with the Hadasim and Arabot), recite the Beracha, and then pick up the Etrog and perform the Na’anu’im (waving) with the four species. This is, indeed, the view of the Shulhan Aruch.
Others, however, maintained that one should first take hold of the Etrog, before the Lulab. The work Emek Beracha, written by the father of the Shela Ha’kadosh (Rav Yeshaya Horowitz, d. 1630), cites Rav Yehoshua Ibn Shueb, a disciple of the Rashba (Rav Shlomo Ben Aderet of Barcelona, 1235-1310), as holding this opinion. Ibn Shueb also maintained that after one fulfills the Misva, he should first put down the Lulab before putting down the Etrog. He compares the Lulab and Etrog in this regard to the Misva of Tefillin, which requires having the Tefillin Shel Yad on one’s arm whenever the Tefillin Shel Rosh is on one’s head. Just as one first wears the Shel Yad before the Shel Rosh, and removes the Shel Rosh before the Shel Yad, likewise, according to Ibn Shueb, one should take the Etrog before the Lulab, and put down the Lulab before the Etrog. Ibn Shueb notes that when the Torah presents the Misva of the four species, it mentions the Etrog first, before the other three species, indicating that it should be taken first. The Emek Beracha writes that it is proper to follow the view of Ibn Shueb, because "all his words are words of authentic tradition."
This was also the opinion of the Magen Abraham (Rav Abraham Gombiner, 1637-1682), in his commentary to the Shulhan Aruch.
Moreover, Hacham Abraham Hamoui of Aram Soba (Aleppo), in his work Bet Simha (Laws of Lulab, 3), writes that one should first take the Etrog, recite the Beracha, and then take the Lulab.
Hacham Bension Abba Shaul (Israel, 1924-1998), in Ner Sion, writes that this is, indeed, the preferable practice. He notes that although the Shulhan Aruch writes that one should first take the Lulab, there are those who understood that the Shulhan Aruch’s intent was not to establish this as the sequence that must be followed. Given the large number of Halachic authorities who felt it more proper to take the Etrog first, Hacham Bension felt that this is preferable.
This was also the opinion of the Hatam Sofer (Rav Moshe Sofer of Pressburg, 1762-1839), who added that the Etrog should be held right side up when it is taken before the Beracha. The Hatam Sofer opposed the practice observed by some to first hold the Etrog upside-down, noting that this is akin to wearing Tefillin on the wrong arm, and thus meaningless.
Summary: Although the common practice is to first pick up the Lulab, recite the Beracha, and then pick up the Etrog, the preferable practice is to first pick up the Etrog, and then recite the Beracha and pick up the Lulab. After one fulfills the Misva, he should put down the Lulab and then put down the Etrog.
