Why does the Torah insert the Mitzvah of Shabbos here?
Hadar Zekenim: After Hashem commanded to honor only Hashem, in any case you should honor Shabbos, for this testifies that Hashem created the world in six days and rested on the seventh.
What constitutes the Mitzvah of "Zachor ... "?
Ramban #1 (citing the Mechilta): It is a Mitzvah to remember the Shabbos every day, so as not to confuse it with another day when it arrives. Because by remembering Shabbos at all times, one thinks constantly about the Creation, thereby acknowledging that there is a Creator. 3
Ramban #2 (citing the Mechilta): It constitutes calling the days of the week after the Shabbos (Echad be'Shabbos, Sheini be'Shabbos ... ). 4
Seforno: It is an injunction to remember Shabbos whilst one is occupied with one's affairs during the week, so that when Shabbos arrives, one is able to sanctify it by pushing all one's mundane activities out of one's mind. 5
Rashbam: It is a Mitzvah to remember the first Shabbos of the Creation. 6
Just as Hashem sanctified Shabbos by the Manna not falling on Shabbos, but on Friday (Refer to Bereishis, 2:3:1:1).
See Ramban, who maintains that this is the individual opinion of Shamai, but is not Halachah.
See Ramban DH 'u'le'Kach Amru', who elaborates on this point.
Why does the Torah write "Zachor" here, and "Shamor" in the Aseres ha'Dibros in Va'eschanan?
Rashi, from Rosh Hashanah (27a): In fact, "Zachor" and "Shamor" were said simultaneously and heard simultaneously. 1
Hadar Zekenim: Zachor applies to something ancient (Ma'ase Bereishis). Shamor applies to something recent (Yetzi'as Mitzrayim).
Hadar Zekenim and Riva, both citing a Midrash: A parable for this is one who sent his son to buy oil; the son lost the money and broke the flask. The father gave to him more money and a new flask, and scratched him. He said, remember that I scratched you, and guard the money and flask! Also here, after the Mekoshesh was killed for Chilul Shabbos, it says Shamor. 2
What is the difference between "Zachor" and "Shamor"?
Ramban #2 (according to Kabalah): Zachor refers to the day of Shabbos, and Shamor to the night. 3
Ramban #3: Zachor is a Mitzvah to be Mekadesh it at the beginning. 4 Shamor is not to do Melachah.
As the Gemara says in Eruvin (96a) 'Hishamer, Pen and Al are all Mitzvos Lo Sa'aseh' (Ramban).
Mitzvos Asei are rooted in Ahavah and Mitzvos Lo Sa'aseh in Yir'ah, and it is because Ahavah is greater than Yir'ah that an Asei generally overrides a Lo Sa'aseh (Ramban - See Rav Chavel's commentary on the Ramban who elaborates at length on this point).
Which explains why we say 'Bo'i Likras Shabbos Malk'sa ... ' on Friday night, and why, on the other hand, the Gemara in Pesachim, 106a refers to the Shabbos morning Kidush as Kidusha Raba (Refer to 20:8:4:1**).
Moshav Zekenim: Some say that mid'Oraisa, Kidush must be over wine. And some say that it is only mid'Rabanan.
What did Hashem achieve by adding Shamor?
Ramban: Had the Torah written only "Zachor", women would be Patur. 1 Now that Hashem added "Shamor", they are obligated to keep Shabbos just like men.
As they are from all Mitzos Asei that are time-related (Ramban).
What are the ramifications of "Le'kadsho"?
Ramban #3 (the Mechilta): It is a Mitzvah to sanctify the Shabbos by reciting Kidush over wine 3 when it arrives.
Ramban: To spend the day visiting the Chachamim and the Nevi'im, in order to hear the word of Hashem. And, commenting on the Gemara in Chulin, 5a, which states that like Avodah-Zarah, Shabbos is equal to all the other Mitzvos, he explains that it is because on Shabbos, we attest to all the major components of Emunah - Creation, Supervision and Prophecy.
See Ramban DH 'Aval le'Raboseinu' and 'uvi'Gemara Pesachim'.
Rashi, from Rosh Hashanah (27a), writes that "Zachor" and "Shamor" were said at once. All the Aseres ha'Dibros were said at once!
Moshav Zekenim: When each of the Dibros was said by itself, "Zachor" and "Shamor" were said at once.
Moshav Zekenim #2: When Hashem explained Aseres ha'Dibros to Moshe, and [Moshe] said Zachor, He immediately said Shamor.
Rashi writes that other seemingly contradictory pairs of Pesukim were said at once. "Zachor" and "Shamor" do not contradict each other! (Why did they need to be said at once?)
Moshav Zekenim #1: "Zachor" is to be Mekadesh Shabbos and not do Melachah, like it says in the continuation. In Devarim, it says "like I commanded you." It did not need to give the reason [that Hashem created the world in six days].
Moshav Zekenim #2: Here, it mentions creation, and does not command that your slaves rest. In Devarim, it commands that your slaves rest, for you were a slave and Hashem freed you. They needed to be said at once to connect them. In Kidush, we say Zecher l'Ma'ase Bereishis corresponding to Zachor, and Zecher l'Yetzi'as Mitzrayim corresponding to Shamor.
Hadar Zekenim: "Mechaleleha Mos Yumas" corresponds to Shamor, and "uv'Yom ha'Shabbos Shnei Chevasim" (Korban Musaf of Shabbos) to Zachor. They contradict each other, so they needed to be said at once.