First of all, what do you mean by emotions one is "supposed to feel?" Doesn't a person just feel one way or another? How can one be commanded to control one's emotions?
Good question(s)! But in Judaism, the approach is that thoughts, and even one's emotions are, to some extent, under one's control.
As proof, the Torah requires the Jew to love, on certain occasions, as in "Love your neighbor as yourself" (Vayikra, 19:18), or "You shall love him (the stranger) because you yourselves were strangers in the Land of Egypt; I am the L-rd your G-d" (Vayikra, 19:34). Perhaps most important is the command to love G-d, "And you shall love the L-rd your G-d with all your heart, and with your whole life, and with all your strength" (Devarim, 4:5).
And also not to hate, as in "Do not hate your neighbor in your heart" (Vayikra, 19:17), with the possible exception of Amalek, the nation whose impudent and cowardly attacks upon the weakest members of the Children of Israel when they left Egypt showed that they feared man but not G-d. We are commanded to remember this and struggle against their particular brand of evil in every generation (Devarim, 25:17-19), even as G-d Himself, so to speak, wages an Eternal Battle against them (Shemot, 17:16).
Having established the legitimacy of the question, the emotion which we are supposed to feel is a kind of partial mourning, with certain restrictions on behavior, which are lifted on Lag BaOmer, the Thirty-Third Day of the Omer.
Why mourning? What should we be sad about?
There are at least four answers, all related:
One is for the lost Bait HaMikdash, the Temple, which was considered both the "residence," so to speak, of G-d on earth, and the central symbol of the Jewish Religion, a situation which has existed for more than nineteen hundred years.
Another is for the tragic loss by the Jewish People of the Military Campaign which centered on the City of Beitar. This struggle pitted Jewish forces led by a great Jewish leader by the name of Bar Kosiva, against the most powerful of the Roman legions, led by the ablest of the Roman generals. This uprising of the Jewish People in approximately 135 C.E., which by archaeological evidence definitely included the minting of coins, and which some have speculated included the re-conquest of Yerushalayim, with at least the beginning of work on Temple Reconstruction, came to a tragic end with the slaughter of hundreds of thousands of Jews.
Another source of tragedy is the failure of the man whom Rabbi Akiva, one of the greatest Jewish leaders of all time, had declared to be the Mashiach. This man, Bar Kosiva, was given the name "Bar Kochba," "Son of the Star," based on a verse of Messianic Prophecy uttered, paradoxically, by one of our People's greatest Biblical enemies, Bilaam the Seer. "I see him, but not now; I perceive him, but not in the immediate future; a star will shoot forth from Yaakov, and a Tribe will arise from Israel, which will destroy the ends of Moav and uproot all the Children of Seth."
A fourth reason for sadness, the traditional one, is based on the reference in the Talmud (Yevamot, 62B). "It was said, 'Rabbi Akiva had twelve thousand pairs of students, from Gevat to Antipatros, and they all died in one short period of time, because they did not have proper respect for each other.' And we learned in a Tosefta, 'They all died between Pesach and Shavuot.' " The deaths from plague ceased completely on the Thirty-Third day of the Omer (Lag BaOmer).
Now the reason given for the plague is very strange, considering the fact that their great teacher had proclaimed that the essence of the Torah is "You shall love your neighbor as yourself." (Vayikra, 19:18) Ethical behavior between Man and Man and between Man and G-d can be said to be the main "Purpose" of the Torah. How then could his students in large numbers have ignored their master's basic teaching?
One possibility is that the students were actually perfect in regard to this trait. But the People of Israel did not have proper respect for each other. The students tried to "take the blame" for the People as a whole, even as Moshe tried to do, according to one of the commentators, for the People of Israel, after they constructed the Golden Calf.
It is also possible that the recounting of the information about the plague was written for the eyes of the Roman and later, Christian, censors of the Talmud. These would not, presumably, have looked kindly at twenty-four thousand devoted students of the Torah who had been leaders of a rebellion, and who had perished heroically in battle.
Another extremely sad chapter in our history, associated with the Period of Sefirat HaOmer, were the Massacres of the First Crusades which occurred in Germany in 1096, in which many Jewish communities were utterly destroyed, some burned to the ground, with many great Torah scholars perishing "Al Kiddush Hashem," "for the Sanctification of the Name of G-d."
Tragedy followed tragedy, again occurring primarily during the Period of Sefirat HaOmer, during the Bogdan Chmielnicki Massacres in Russia and Poland, in the years 1648 and 1649, in which some three hundred thousand Jews were massacred, with a cruelty and ferocity rarely seen before.