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.

Kiddushin Daf 3

The Gemara describes an Esrog as 'darko legadel al kol mayim' it grows from all water. There is a well known idea that the four species represent four different types of Jews that we take and bring together. The Esrog has taste and smell, wisdom and good deeds, the Lulav just taste, the Hadasim just smell, and the Aravos neither. Where did the Esrog get to have both taste and smell? The answer is because the Esrog knows what is the proper way to grow, from all water. The Mishna in Avos teaches us "aizehu chacham, halomed mikol adam' Who is a wise man? One who learns from everyone. Water is compared to Torah and the Esrog is willing to grow from all water, wherever the Esrog can find something to grow from he will take it in and use it for personal growth. We too should strive to be like an Esrog and look for possibilities for growth from everyone we encounter.

Kiddushin Daf 2

The Gemara tells us that the word 'derech' way can be both masculine or feminine depending on what it's referring to. Therefore in the Pasuk 'vihodatam lahem es haderech yailchu bah' the use of the word derech is feminine since it refers to the Torah which is feminine as we see in the verse 'Toras Hashem temimah mishivas nafesh'. The question is why is the Torah a feminine word and not a masculine word. Perhaps the answer is that the Torah requires our effort to attain it. Just like a man has the role to seek out a woman, so too, a man has to seek out the Torah. People expect to wake up one day knowing all there is to know. The Torah needs to be acquired as delineated by Rabbi Meir in Pirkei Avos. There are forty eight ways one must utilize in acquiring Torah it doesn't happen by itself!

Gittin Daf 47

When Reish Lakish died, he left over some saffron to his heirs. He (regretted that he worked more than he needed to, and) applied to himself "they left their wealth to others." We can ask what is so terrible about leaving over an inheritance to our children??? We can learn a valuable lesson from Reish Lakish. We spend much of our time on physical pursuits and neglect our spiritual ones, Reish Lakish is teaching us that the legacy that we leave to our children is not the amount of money they inherit but the lessons on life that we show them by our example. If we spend our days and nights at work and neglect Torah study, every extra cent we earned that we didn't use at the end of our life is wasted time that we could have put to better use.

Gittin Daf 46

When a woman was divorced because she was thought to be an "ailoniss" (a woman who cannot conceive)R' Yehuda says the husband cannot remarry her after the divorce. If the woman marries another and conceives and wants to claim her Kesuba from her first husband, R' Yehuda says we tell her "shtikutech yapheh midiburech" your silence is better than your speaking. This is because if she claims her Kesuba, the first husband might say that he would not have divorced her had he known she could conceive and that would question the validity of the Get and also her children.
This advice of R' Yehuda is something we all should take to heart. Many times our speech brings about more problems for us than gain.