Gemara

6b (R. Levi bar Chiya): If one leaves a Beis ha'Keneses and goes to a Beis Midrash to learn Torah, he merits to greet the Shechinah -- "Yelchu me'Chayil El Chayil Yera'eh El Elokim b'Tziyon."

(R. Chelbo): One who leaves a Beis ha'Keneses should not take big steps.

(Abaye): This applies only to leaving, but it is a Mitzvah to run to a Beis ha'Keneses -- "Nirdefah Lada'as Es Hash-m."

(R. Zeira): At first, when I saw Talmidim running to the Kalah (public lecture on the Shabbos before the festival), I thought that they were disgracing Shabbos;

Later, I heard R. Tanchum teach that one should run to learn, even on Shabbos -- "Acharei Hash-m Yelchu k'Aryeh Yish'ag." Now, also I run.

Shabbos 113a (Beraisa): "Me'Asos Derachecha" - you should not walk on Shabbos the same way you walk on a weekday;

113b - Question: How should one walk differently on Shabbos?

Answer #1: Rav Huna taught that if one encounters a stream, if he can leave one foot on the ground and step across, it is permitted [even to jump]. If not, he may not [jump across].

Objection (Rava): Should he rather go around? This causes him extra walking! If he will walk through the water, his garments may get soaked and he will come to squeeze out water!

Rather, since there is no alternative, he may jump across.

Answer #2 - Question (Rebbi): May one take a big step on Shabbos?

Answer (R. Yishmael b'Rebbi Yosi): One may not even during the week, for it diminishes one part in 500 of his sight. It is restored at Kidush of Shabbos night.

Eruvin 42a (Rav Nachman): If one was walking and does not know where the Techum ends, he takes 2000 medium steps. This is the Techum.

Rishonim

Rif and Rosh (3b and 1:7): It is a Mitzvah to run to a Mitzvah, even on Shabbos.

Ma'adanei Yom Tov (300): This is the Rif's text also in Shabbos. However, our text, and the Rosh's text in Shabbos, says 'to Devar Halachah.' An advantage of this text is that it explains why R. Zeira did not bring the verse "v'Nede'ah Nirdefah...", which precedes the verse he brought "Acharei Hash-m Yelchu." The verse he brought does not explicitly discuss (rapid) pursuit! According to the text "l'Devar Halachah", we could not have brought "v'Nede'ah Nirdefah...", for not everyone knows that the reward for the Shi'ur is running to it!

Ha'Kosev (16b in Sefer Ein Yakov): Why did he need to say 'this applies only to leaving?' Rav Huna discussed only leaving! I answer that one might have said that we learn from the reasoning. Perhaps similarly or all the more so one may not run to the Beis ha'Keneses. If one who already prayed slowly may not run out, all the more so if one runs to pray it looks like he seeks to exempt himself from a burden. Logic would have said so, if not for the verse.

Ma'adanei Yom Tov (200, on Rosh 1:7): The Ein Yakov answered (I found this answer in ha'Rif, but not in ha'Kosev - PF) answered that we forbid leaving with big steps only if he is not intending to return. If he intends to return, he should leave with big steps, to enable him to return quickly. I lean to be stringent about this, for one who sees him does not know that he intends to return. To him, the big steps look disgraceful, like one who rushes to leave. Also, his Perush is only according to texts that say Lemeipak Lemei'al (to leave in order to come [back]. The Rosh's text "bede'Nafik" does not connote that he left without intent to return. I say that one might have thought Rav Huna taught a Diyuk: one may not leave with big steps, but he may enter with big steps. Abaye teaches that this is wrong. It is a Mitzvah to run to enter!

Rambam (Hilchos Tefilah 8:2): It is a Mitzvah to run to the Beis ha'Keneses. When he leaves, he may not take big steps. Rather, he walks slowly.

Poskim

Shulchan Aruch (OC 90:12): It is a Mitzvah to run to the Beis ha'Keneses, and to any Mitzvah, even on Shabbos when it is forbidden to take big steps.

Kaf ha'Chayim (86): The Prishah brings from R. Yonah (Sefer ha'Yirah) that the Mitzvah is only when he is close. Keneses ha'Gedolah says that the Mitzvah is from when he leaves his house. R. Yonah means that when he is close, he runs faster. This seems primary. Running is not primarily for those who see him, rather, for him himself to do the Mitzvah with zeal. Yefeh l'Lev says that one should not run in a street of Nochrim, lest they mock him. If so, if one merely goes quickly without running, there is no mockery, so he does so even in a street of Nochrim.

Magen Avraham (24): it is a Mitzvah to run until the entrance of the Beis ha'Keneses. One may not run in the Beis ha'Keneses. Rather, he walks with fear.

Magen Avraham (25): Even during the week, taking big steps removes one part in 500 of a person's sight.

Eshel Avraham: One may not take a big step even when going to a Mitzvah. When damage is common (we do not say that one who does a Mitzvah will not be harmed). The Mitzvah to run is without taking big steps, unless he normally takes big steps and cannot refrain. On Shabbos he must force himself to refrain, but it is permitted when going to the Beis ha'Keneses. The Maharsha connotes that it is a Mitzvah to take big steps (when heading to Mitzvos). This requires investigation.

Mishnah Brurah (40): All the more so one must be careful not to stop on the way to converse with someone about his own desires. Often, due to this one comes after Borchu and Kedushah. The Prishah says that the primary running is when he is close to the Beis ha'Keneses, for then it is clear that he runs for the Mitzvah of (going to) the Beis ha'Keneses. The Pri Megadim says that in the morning, when he goes holding Talis and Tefilin, it is always recognized.

Shulchan Aruch (ibid.): However, it is forbidden to run when leaving the Beis ha'Keneses.

Magen Avraham (26): If he leaves in order to return, it is a Mitzvah to run, so that he can return quickly. I say that there is no concern for people who see him leave. As long as the entire Tzibur did not leave the Beis ha'Keneses, presumably he intends to return. Also, it is better that they suspect him, than for him to neglect hearing Kaddish or Kedushah. Also the Shlah says that one who leaves the Beis ha'Keneses to go to the Beis Midrash should run (even though people might not know that he is going to learn - PF).

Eshel Avraham: This is when the Beis Midrash is near, so it is evident that he runs to it. One may not run (from the Beis ha'Keneses, to his pursuits) even without big steps.

Mishnah Berurah (43): It is forbidden to leave running or with big steps only if he goes to his pursuits. Then it seems that being in the Beis ha'Keneses was a burden to him.

Kaf ha'Chayim (90): It is not proper to push to leave in front of others, and surely not to leave while saying Aleinu or before the end of the Tefilah.

Shulchan Aruch (301:1): One may not run on Shabbos, unless it is to do a Mitzvah, e.g. to the Beis ha'Keneses or a similar matter.

Magen Avraham (1): Even during the week one may not take big steps, for it diminishes sight.

Kaf ha'Chayim (2): He did not explain what is different about Shabbos. The Maharsha says that during the week it is permitted, just it is harmful. On Shabbos it is forbidden. Machatzis ha'Shekel says that a blind person is forbidden only on Shabbos. Tosefes Shabbos says that the Halachah does not follow the opinion that it harms the sight. It is forbidden only on Shabbos.

Rema: One may not take a step bigger than one Amah, if he can take a smaller step.

Gra (DH Im): We learn from Shabbos that when it is impossible (not to take a bigger step), it is permitted.

Mishnah Berurah (2): Likewise, one may not jump on Shabbos.

Magen Avraham (2): One should not step more than a half Amah in front of the other foot. The foot itself is a half-Amah. The Agudah brings from Tosfos that one should not remove a foot from the ground before the other foot rest on the ground.

Gra (DH v'Asur): We learn from Eruvin 42a that a normal step is one Amah. Also Rashi in Shabbos says so.

Mishnah Berurah (3): The Shi'ur for a bigger person is according to the size of his steps. (Presumably, it is twice the length of his foot - PF.)

Torah Lishmah (40, attributed to the Ben Ish Chai): One should not ride to the Beis ha'Keneses just for the convenience, for he loses the Mitzvah to run.

Tzitz Eliezer (12:17) and Yalkut Yosef (155:1): If one must drive, he should park some distance away, to fulfill the Mitzvah of running to the Beis ha'Keneses.