INTRODUCTION: THE OBLIGATION TO BE HAPPY ON YOM TOV
All Jewish adults are biblically obligated to be happy throughout the three festivals: Pesach, Shavuos, and Sukkos (Mishna Berrura 529:15). As the Passuk says (Devarim 16:14-15) 'Rejoice in your festivals … and you should be exceedingly happy', this is known as the obligation of Simchas Yom Tov - to be happy on Yom Tov!
In times of the Beis Hamikdash, the joy of these pilgrimage festivals was expressed primarily through bringing the Chagigah offerings in Yerushalayim (Devarim 27:7).
Since the destruction, the Talmud (Pesachim 109a) rules that men fulfill this mitzvah by drinking wine and having a festive meal. In addition, whilst not obligatory according to most, it is a mitzvah to have meat (as clarified by Biur Halacha 529:2 and Magen Avraham 696:15, unlike Rambam 6:18 who mandates it). Women should be bought new clothes and jewelry as one's financial means allow and children should be given 'roasted seeds, nuts, and sweets' (Rambam Hilchos Yom Tov 6:17, Shulchan Aruch OC 529:1).
THE OBLIGATION TO DRINK WINE
According to some opinions, in order to fulfill the obligation of wine on Yom Tov, one must drink at least 2.92 fl oz / 86 ml of wine and if possible 5.07 fl oz / 150 ml (these are the Revi'is measurements of Reb Chaim Na'eh and the Chazon Ish respectively). But others say even a small amount will suffice (Chut Shani, Yom Tov 22:1 from Reb Nissim Karelitz and Leket Hilchos Yom Tov 1:1 from Rav Shmuel Vozner and Reb Chaim Kanievsky).
The Magen Avraham (MA 99:1) rules that since drinking in a Yom Tov meal will impair one's concentration, a Torah scholar may not subsequently give Halachic rulings as they are considered impaired. Some Torah giants would drink so much wine during the Yom Tov meals, that they refrained from giving rulings until the following day (Beitza 4a, Kerisus 13b, Shach YD 242:19).
According to most opinions, one can not fulfill this obligation with other alcoholic drinks (Taz OC 552:1, Reb Moshe Feinstein quoted in Hilchos Chol Hamoed Zichron Shlomo, but Rashi Bava Metziah 64b indicates otherwise).
Drinking grape juice does not fulfill the mitzvah, as it is not alcoholic, it is not a mood enhancer. If one enjoys it, according to some it suffices (Shaagas Aryeh 65, Rav Moshe Feinstein, quoted in Chol HaMoed; Kovetz Teshuvos 1).
THE MITZVAH TO EAT MEAT
The Mitzvah to eat meat on Yom Tov (whilst not an obligation as detailed above), is defined as animal meat as opposed to poultry (Rashi Chagiga 7b, Magen Avraham OC 551:28, Darkei Teshuva YD 89:19). However, a minority opinion maintains that nowadays poultry also suffices (Yad Efraim YD 1:1).
WHAT IF I DON'T ENJOY WINE/MEAT?
If a person does not enjoy drinking wine or eating meat, he does not have to force himself to do so. Rather, for the Yom Tov meals, he should make sure to have the foods which make him happiest (Shaagas Aryeh 65).
It should be noted that the Poskim write that independent of the mitzvah of Simchas Yom Tov, there is an obligation to honor Yom Tov the same way one honors Shabbos. Therefore, regardless of what is served at the Yom Tov meal, the meal should be held to the same delicacy standards as that of a Shabbos meal (Rambam Ibid 6:16 and Shulchan Aruch Ibid)
There are three opinions regarding the frequency of the mitzvah to eat meat on Yom Tov:
The Shulchan Aruch (Ibid) writes that one should drink wine at every Yom Tov meal which implies that the mitzvah of Simchas Yom Tov applies at every meal. This also seems to be the opinion of Darkei Teshuva (89:19), who questions the custom to eat dairy products on Shavuos based on the mitzvah to eat meat as part of Simchas Yom Tov (more on this below).
Rav Moshe Feinstein (Igros Moshe Orach Chaim 3:68), notes that the mitzvah of eating meat is patterned after the obligation to eat the Chagigah sacrifice. Just as that applies once a day, for every day of the holiday (including Chol HaMoed), so too nowadays the mitzvah to eat meat applies once every day.
R. Moshe Shternbuch (Moadim Uzemanim 1:29), argues that there is no set frequency for the mitzvah of being happy on Yom Tov. There is simply a constant obligation to be in a state of happiness. One uses certain mediums to achieve that state, and when the effect of those mediums wears off, one must replenish through those mediums!
THE SHAVUOS CONUNDRUM!
Given that there is a mitzvah to eat meat on Yom Tov many wonder how to reconcile this with the prevalent custom to eat milky meals on Shavuos. Listed and explained below are the four prominent customs:
Some eat only dairy the entire Shavuos in order to fulfill the minhag but many disapprove of this practice for the above reason (Darkei Teshuva 89:19).
The most widely accepted Ashkenazic custom is to split up the meal and eat dairy and then after washing their mouths out with a liquid and food and washing their hands eat meat (Remah OC 494:3, Magen Avraham, Mishna Brurah, Aruch Hashulchan, and Chayey Adam Ibid).
Some families eat dairy for the nighttime meal and meat for the daytime meal. Such was the minhag of some scholars (Steipler Gaon in Orchot Rabbeinu Vol.2 p.98). This likely follows the opinions that on the night of Yom Tov, the obligation of Simchas Yom Tov is rabbinical rather than biblical (Shaagas Aryeh 68 which Mishna Berurah in Shaar HaTziun OC 546:15 relies on).
The Kabbalah-oriented who follow the Zohar (Parshas Mishpatim 155a) that doesn't allow milk and meat in one meal or in the same hour, either a) eat a dairy meal, conclude with Birkas Hamazon, and then a meat meal or b) as is the Chassidic custom, eat a dairy Kiddush on Shavuos morning (which must include 'Mezonos'), having a regular Yom Tov meal with meat later (Pri Megadim M"Z 89:3, Darchei Teshuva YD 89:19).