Succah Daf 12-13

Abaya says one may use שושי for the סכך but not שווצרי This is because שווצרי leaves a bad odor and one will then feel compelled to leave the סוכה. We see from this a very important lesson- even if our behavior fulfills the requirements within the exact letter of the law, if we leave a bad odor- i.e. we cause people to not want to be around us- we are not acting validly.

Pesachim Daf 42

Rabbi Yehuda teaches that one must use מים שלנו (water that drawn before nightfall) for the preparing of  מצה. Rav Masna taught this law to the people of Papuna and the next day they came to Rav Masna's house with their containers asking for Rav Masna's water. This is because they assumed the meaning of  מים שלנו was "our water".
The lesson here is the amazing faith in the Rabbi's words that the people of Papuna displayed. They did not even question the requirement to use only the water of Rav Masna. If he said that was what was required, that's enough for them. How often do we discount the words of our Sages without even giving it the benefit of the doubt, let alone accept what we our told to do faithfully?

Pesachim Daf 34

Reish Lakish teaches us that with regards to Holy things "היסח הדעת" inattentiveness renders the Holy thing unfit. Even if we know that it did not become טמא (ritually impure) the lack of guarding the Holy thing from  becoming impure is already enough. This can be a powerful lesson for us. There are many things in our lives that can be considered Holy. We ourselves are referred to as a ״גוי קדוש״ a Holy nation. Thus we must be vigilant in not allowing any היסח הדעת to render our sanctity compromised.

Jewish Humor

One of the most famous examples of Jewish humor in the Torah is when the Jewish people were leaving Egypt. They had the Red Sea in front of them and the mighty Egyptians behind them. At that moment the B'nei Yisrael complained " המבלי אין קברים במצרים לקחתנו למות במדבר" "Is it because there is no more room for cemetaries in Egypt that you took us out to die in the desert?"
This comment is extremely funny when you think about it, and in fact Rabbi Samson Raphael Hirsch comments that a sense of humor has always been a way the Jews survived throughout our years of exile. However, I believe that the ability to find humor in this dire situation was in fact the very means to enable a miracle to occur. For if the Jews would have been paralyzed with fright there would indeed be no hope.  Only once the Jews were able to laugh at their predicament did that open the means to salvation.

Pesachim Daf 28

The gemara discusses someone who is tamei or too far away to bring the korban pesach and says this does not mean he should give up on eating matza and marror. This is an important lesson for us because often people think if I can't do it perfect I should skip the whole thing but it is not all or nothing. Often people are hesitant to learn daf yomi because they erroneously think it is not worth it if I will be bound to miss a day here and there, but in reality every day is precious and has boundless value.

Pesachim Daf 27

Rabbi Yehuda is of the opinion that only burning is a valid means to destroy chometz. He bases this on a logical inference (kal vechomer) from nosar ( leftover parts of a Korban) If  nosar which doesn't encompass any ongoing transgressions till its destroyed, needs to be destroyed specifically by fire, surely chometz which has ongoing transgressions (bal yeraeh and bal yematzeh) surely needs to be destroyed specifically by fire. The Chachamim argued "kol din sheatah dan tchilaso lehachmir vesofo lehakel aino din"  whenever one intends to learn a stringency and it ends up a leniency in is not valid.

This is an important message, for in order to be stringent one must always look at the repercussions lest it turn out to be a leniency in the end.

Shavuos 1

Daf 5

The Gemara explains that according to Rebbe "Yedeas Beis Rabbo" is considered knowing. This means that since when a person is young he is taught that one can contract tum'ah in various ways, therefore when he is older and came in contact with tum'ah, he is considered having been aware that he was tam'ei. This is true even if at the time he touched the tum'ah it didn't occur to him that he was tam'ei. This needs explanation. The lesson though, is clear. Things we are taught when we are young really never leave us, and as much as we are unaware now, we can always connect back to our previous knowledge.