Tana Kama's opinion:

The first Shabbos (some translate "week") the mourner should not leave his house.

The second Shabbos (or week) he goes to Shul but does not sit in his usual seat.

The third Shabbos (or week) he can sit in his seat, but he shouldn't talk to anyone.

From the fourth Shabbos (week) and on he is back to normal.

R. Yehudah's opinion

The first and second Shabbos (week) he should not leave his house.

The third Shabbos (week) he goes to Shul but does not sit in his usual seat.

The fourth Shabbos (or week) he can sit in his seat, but he shouldn't talk to anyone.

From the fifth Shabbos (week) and on he is back to normal.

A mourner should not marry during Shloshim.

If the mourning is for one's wife, he should not get remarried until the three Regalim have passed.

R. Yehudah permits getting remarried before the third Regel.

If the mourner (even for his wife) has no children yet he may remarry immediately.

If his wife left him small children who require looking after, he may also marry immediately, like when Yosef Hakohen married his sister-in-law at his wife's funeral.

In Yosef Hakohen's case he nevertheless did not consummate the marriage until after Shloshim.

A Beraisa records three opinions:

(Tana Kama): Freshly pressed clothing, whether new or old, may not be worn during Shloshim.

(Rebbi): Only new pressed clothing is forbidden.

(R. Elazar b'Rebbi Shimon): Only new white pressed clothing is forbidden.

The Halachah:

Abaye conducted himself like Rebbi.

Rava conducted himself like R. Elazar Berebi Shimon.

There is a disagreement between the Judeans and Galileans as to whether any sort of mourning rites are observed on Shabbos (in private only).

Proof (that there is mourning): Because Shabbos counts into the seven days of Shiv'ah.

Proof (that there is no mourning): The Mishnah (above, 19a, 2:c) says Shabbos does not cancel the Shiv'ah. Why would anyone even think that Shabbos cancels the Shiv'ah if mourning is observed on it?

The Gemara offers possible refutations to both proofs.

The Gemara attempts to show that the two opinions are found in a Beraisa. The Beraisa deals with someone who has lost a relative and has not yet buried him. (This status is called in later rabbinic literature Onen, so we will call him that here.) It makes several points:

The Onen may not eat in the presence of the body, but should go elsewhere, or if this is not possible he should construct a partition of ten tefachim, or if this is not possible he should at least turn around while he eats.

The Onen may not eat meat or drink wine.

He may not recite Brachos over food, whether before or after eating.

He is exempt from all positive Mitzvos.

On Shabbos he may eat normally - including meat and wine - and he must recite Brachos, and he must perform all Mitzvos.

Raban Gamliel makes a mysterious statement: Since he must observe these, so must he observe everything. What does Raban Gamliel mean to include that the mourner must do that the Tana Kama did not include? It must be marital relations. (Raban Gamliel says this aspect of Shabbos is also to be observed, while the Tana Kama forbids marital relations.) Hence we see that the issue of private observance of mourning is a disagreement between Raban Gamliel and Tana Kama.

This equation is rejected, because the topic of the Beraisa is an Onen and not a regular mourner. Perhaps the Tana Kama would agree that a regular mourner does not have to refrain from marital relations, or perhaps Raban Gamliel would agree that a regular mourner does have to refrain.