Touching the Mezuzah upon leaving and entering ones home:
There are opinions which hold that when a person leaves his home, he should place his hand on the Mezuzah and say “Hashem Yishmor Tzeisi Uvoi Meyata Viad Olam”. Similarly one should place his hand on the Mezuzah upon entering his house.
Leaving the city: When leaving the city one is to place his hand on the Mezuzah and say the verse of “Bishimcha Tal Atleh” when touching the Mezuzah. [Some say one is to place his middle finger on the Mezuzah.]
Kissing the hand:
The custom today is to touch the Mezuzah upon leaving and entering every room, and to then kiss ones finger [or to kiss ones hand and then touch the Mezuzah]. The Rebbe was not particular to kiss his finger after touching the Mezuzah.
Kissing the Mezuzah before sleep:
It is customary for children to kiss the Mezuzah of their room prior to going to sleep. Parents of all Jewish homes are particular in this matter. This matter applies to adults as well, that they should kiss the Mezuzah prior to sleep. See  rule a Mezuzah which does not have any covering over the parchment [not even a plastic wrap] should not be touched with ones bare hands. It is rather to be touched with a cloth, such as with the use of one’s sleeve and the like. Others however rule that there is no prohibition in touching the parchment. Practically the widespread custom is not to be stringent and people hence touch and hold the Megillah directly without washing hands. Nevertheless it is proper for every person to be stringent upon himself and avoid touching the parchment of a Kosher Megillah [Mezuzah] anytime, unless he washes his hands beforehand, or holds onto it using a cloth.
 Rama 285/2
 Maharil Avoda Zara 11
 The Arizal writes that one should place his middle finger over the name “Sha-kai”. [Birkeiy Yosef]
 “G-d will guard my leaving and returning from now and ever”
 Shach 285/4
 Arizal brought in Matzas Shmurim; Pischeiy Shearim 285/2
 Toras Menachem 1987 2/647; See Likkutei Dibburim Likkut 32 for a fascinating story on this matter
 Kitzur SHU”A 71/4
 Teshuvas R. Akivah Eigar 58; brought in Pischeiy Teshuvah 285/4
The reason: As one is to be stringent not to touch any Holy Scripture with ones bare hands [Rama 147/1 [stringency]; M”A 147/1 [forbidden]; Haeshkol Hilchos Talmud Torah 12; Mor Uketzia 691; Sheilas Yavetz [forbidden, brought in Machazik Bracha 691/3]; Elya Raba 147/1 [stringency] brought in Shaareiy Teshuvah 691/4] and the same would apply to a Mezuzah, as it is included within Kisvey Kodesh. [Pischeiy Teshuvah ibid]
The Michaber 147/1 rules that it is forbidden to directly touch a Sefer Torah without a cloth. The Rama brings an opinion [Aguda and Tosafos] that extends this prohibition to all Kisvei Kodesh. The Rama concludes that the custom is unlike this opinion, although it is proper to be stringent if one did not wash his hands. The M”A ibid explains that in truth the reason that we are not accustomed to be stringent is because this prohibition only applies to Kisvei Kodesh that is written with ink on parchment and since our books are not written in this method it is thus permitted to touch them directly. The conclusion of the Rama that it is proper to be stringent is thus superfluous. However a Kosher Megillah which is written with ink on parchment retains the above restriction from the letter of the law and not just as a stringency. [M”A ibid] The Elya Raba ibid however concludes that the ruling of the Rama “it is proper to be stringent” is actually referring to a Kosher Megillah, and it is hence not required to be stringent from the letter of the law as rules the M”A.
Other opinions: The Taz 286/5 implies that he is not bothered by this issue as he does not mention it in his reasoning’s behind why one should cover the parchment.
 Radbaz 2/771; Shvus Yaakov 11; Panim Meiros 1/76; Shaareiy Teshuvah 691/4; Shev Yaakov 11
 Radbaz 2/771 [“we have never seen anyone stringent in this”]; Panim Meiros 1/76; Shaareiy Teshuvah 691/4
 Shaareiy Teshuvah ibid; Keses Hasofer 19; M”B 147/4; Kaf Hachaim 147/7; Hagahos Chasam Sofer that his teacher Reb Nassan Adler was careful not to touch it directly and so was the custom of the Chasam Sofer himself.