Taking a shower after Mikveh:
From the letter of the law, it is permitted for a man to shower after immersing in the Mikveh, although some are stringent not to do so. However, according to all it is permitted to momentarily go under the shower for a quick rinse which will not exceed three Lugin or water, or if it will not fall on one’s head, or majority of one’s body. Nevertheless, on Erev Shabbos [and Erev Yom Tov] according to all it is better not to do so, in order so one not completely wash off the Mikveh water from his body, as will be explained next.
Drying oneself after Mikveh:
Based on the teachings of the Arizal one should not dry the Mikveh water off his body after immersing. Practically, we are particular to dry ourselves with a towel after immersion and one who desires to follow the directive of the Arizal is to leave some part of his body not dried. It suffices for one to leave his feet wet in order to fulfill this directive.
 See Shabbos 14a; Shulchan Aruch Y.D. 201:75; Or Zarua Nida 338; Igros Moshe Y.D. 2:96; Taharah Kehalacha 22:29-30 [pp. 565-567]; Sheivet Halevy 7:33; Tzitz Eliezer 11:64-15; Piskeiy Teshuvos 88:8; Koveitz Hearos Ubiurim 1078 p. 47 for article of Rav S.D. Levin
Background: In Shabbos 14a the Gemara states that one may not shower his head and majority of his body in water after immersing in the Mikveh. It is disputed as to what type of immersion this decree applied to, as brought in Shulchan Aruch 202:75
 Yaskil Avdi 7:8 [that a 9 Kavim shower is permitted from the letter of the law]; Sheivet Halevy 7:33; Tzitz Eliezer 11:64-15; Piskeiy Teshuvos 88:8; Rav Levin ibid
The reason: 1) As even regarding a Nidda the Rama Yoreh Deah 202:75 rules that only some opinions are stringent and so is the custom. However, others argue on Rama and rule doing so is permitted even by a Nidda. [Gr”a ibid; Darkei Teshuvah 202:332; Lechem Vesimla 122; Birkeiy Yosef 202:75 in name of Rabbeinu Yeshayah 152 and Peri Hadama] 2) Furthermore, Tevilas Keri is valid even through Mayim Sheuvim, and the entire reason that the sages decreed against three lugin of water being poured is because of the invalidation of Mayim Sheuvim. [Koveitz Ohalaei Torah ibid] 3) Furthermore, Tevilas Keri is valid even through nine Kavim of a shower and hence certainly it can’t be forbidden to shower after Tevilas Ezra. [See Yaskil Avdi 7:8]
Other opinions: Some Poskim rule one may not shower after Mikveh, just as is the custom regarding a Mikveh. [Tiferes Adam 1:28]
 Yaskil Avdi 7:8 in Hashmatos that a) The world is accustomed to be stringent and b) This is because if one pours less than nine Kavim he is impure; Shevet Halevi ibid concludes that it is proper to be stringent; Piskeiy Teshuvos 260:1 in name of Sheivet Halevy that so is the custom of the world to avoid doing so despite it being allowed from the letter of the law. See also Tiferes Adam 1:28 who rules one should never shower after Mikveh; Rav Levin ibid concludes it is a Hiddur to be stringent as nevertheless a Kosher immersion is done also for Tosefes Taharah, by which perhaps the decree of three Lugin applies
 See Or Zarua ibid that the prohibition is only against bathing and pouring three Lugin of water on her body. Accordingly, it would be permitted according to all for her to enter the shower for a short amount of time, less than three Lugin.
 Taharah Kehalacha ibid based on Gemara Shabbos ibid
 Based on Kaf Hachaim 260:8
 As the water is considered holy with the spirit of Shabbos [and Yom Tov] and it is thus proper for the body to absorb it. [ibid] However see Kaf Hachaim 260:8 which writes that this only applies if one did not immerse more than once, otherwise he has already washed away the main Mikveh water. Nevertheless, he concludes that even these second waters of the Mikveh has some holiness and is thus not to be dried.
 The holiness of that leftover water from the Mikveh will then subsequently spread to the rest of the body. [Rebbe Shaareiy Halacha Uminhag 1:131]
 Shaareiy Halacha Uminhag 1:131