| Pesachim
One of the basic ways we express gratitude is simply to say thank you. One way in which observant Jews do such is by reciting abracha before deriving benefit from this world.
Our tradition teaches that a Jew should say at least 100 brachot a day. At times, we must make two or more brachot simultaneously, and Jewish law had to determine which would take precedence. Beit Shammai and Beit Hillel, for example, argue whether, when makinghavdalah, we should first say the bracha on the besamim, spices, or on the eish, fire (Brachot 52b). Better known is their dispute regarding the order of brachot of kiddush. Beit Shammai are of the opinion that we first say the beracha mekadesh HaShabbat, and only afterward say the boreh pri hagafen on the wine. We, of course, follow the practice of Beit Hillel and first say the boreh pri hagafen(ibid 51b and Pesachim 114a). The principle tadir v'sheino tadir tadir kodem (Zevachim 89a), that which is more common is to be said first, is, at least in this instance, decisive.
In those instances where there are two or more brachot to be said, one can be certain that there will be more than three opinions. When Yom Tov falls on Saturday night, the Talmud (Pesachim 102b-103a) has no less than nine opinions as to the proper order of the five brachot[1] that are said during the "kiddush". While our practice is to first say kiddush for Yom Tov and then make havdalah for Shabbat, this was not universally agreed upon. There is compelling logic that one ought to make havdalah first--we should mark the end of the first day before we mark the onset of the next. The Talmud goes even further, explaining that when we have a king (Shabbat) and a governor (Yom Tov), we should first escort the king, and only afterwards welcome the governor (Pesachim 103a). This has little to do with the order of the days, and everything to do with the hierarchal order of Shabbat vis a vis Yom Tov. And if Shabbat arrives as Yom Tov is ending, we dispense with havdalah altogether.
Yet as we know, that is not our practice. We first say kiddush, and only afterwards do we recite havdalah. As the Rashbam explains[2], sanctifying a day (i.e., kiddush) is "better" than havdalah. Alternatively, he explains, if one were to make havdalah first, it might imply that Shabbat is a burden, and that we are trying to quickly rid ourselves of it.
Amongst the views cited are two that place kiddush and havdalah as the first and last brachot, separated by the other brachot. With kiddush and havdalah representing polar opposites, sanctifying the day and marking the end of such a day should be separated as much as possible.
Yet, ironically, Judaism defines something as kadosh when it is separated and consecrated for a special purpose. The biblical call to be holy is a call to distinctiveness; whether separating from sexual immorality (Rashi, Vayikra 19:2) or eschewing those activities that might be technically permissible, but are corrosive (Ramban). In order to make kiddush, we must make havdalah. Yet it is kedushahthat is our ultimate goal, and it is kiddush that must preceed havdalah.
[1] While it has been a while since I studied statistics, considering that there are 120 possibilities as to the possible order of fivebrachot, we should be relieved to find "only" nine opinions. Even if we ignore the placement of the shehechiyanu, which most of the opinions do, 15 of 24 possible combinations have no voice in rabbinic literature
[2] There are scattered places in the Talmud where it is Rashbam's, and not Rashi's, commentary that is on the inside of the Talmudic page; most notably beginning on Bava Batra 29a, where we find a note, "Here Rashi died, the rest of the commentary is by [his grandson] the Rashbam". However, the only place--to my knowledge--in the Talmud where both appear together is the tenth chapter of Pesachim. I hope some of the scholars on this list can enlighten me as to why.