In the past the Sages said that women should not study the Oral Torah (e.g., Talmud). This was due to the fact that women did not receive any formal education. Nowadays, when women receive a general education comparable to men, they should also study the Oral Torah. This change is related to the fact that the final redemption is drawing closer.

Women learn the halakhot applicable to them, and the teachings of Hasidism are included in this:

First of all: “Women are obligated to learn the halakhot applicable to them in order to know them, such as…all positive mitzvot that are not time-bound, and all prohibitions from Torah law or rabbinic law, in which they are obligated like men” (Shulĥan Arukh HaRav, Hilkhot Talmud Torah 1:14). This includes the vast majority of the Torah’s laws; if only all the men would be fluent in all of this. Another important principle is with regard to studying the inner layers of the Torah through hasidic thought, where all matters of belief in God and His unity, and love and fear of God are explained, as the verse states: “Know the God of your father and serve Him with a whole heart” (I Chronicles 28:9). This study is pertinent to fulfilling the “six commandments whose obligation is constant and does not lapse for even a moment during a person’s entire life” (Sefer HaĤinnukh, Introduction). Women are obligated in them just as men are.

Nowadays women have to study the Oral Torah as well:

Even with regard to studying the Oral Torah in general (in addition to studying the laws pertinent to them), since women and girls study various subjects as it is…they are not only permitted to learn the Oral Torah, but it goes much further. Based on the very reason for this law, it is necessary to teach them the Oral Torah, and not only halakhic rulings without reasons, but also the reasons for the halakhot and even the give-and-take dialogue of the Oral Torah. It is natural (for a man or a woman) to desire and enjoy this kind of learning, and in this way their intellectual development (“cleverness”) will be in the spirit of our holy Torah…. It can be said that the reason we have merited an increase in Torah study for women specifically in recent generations is that preparations for the time of redemption are accelerated at the end of the age of exile. For “in those days knowledge, wisdom, and truth will increase” (Mishneh Torah, Hilkhot Teshuva 9:2). (Rabbi Menaĥem Mendel Schneersohn, Hitva’aduyot 5750, vol. 3, p. 172)

Further reading: For more on every individual’s obligation to learn Torah, see A Concise Guide to Halakha, p. 509.

When the great Rabbi Bunim of Zikhlin came to Rabbi Menaĥem Mendel of Kotzk, he was immediately swept up with the hasidim’s ecstatic service of God, and his diligence in Torah study declined somewhat. The Rebbe noticed and addressed him with the following question: “They say that a learned scholar is one who has filled his belly with the Talmud and the Codes. Why do they say “belly” instead of “head”?

Our nation exists only because of its [Written and Oral] Torahs. Since the Creator said that the nation will remain as long as heaven and earth continue to exist, by definition its Torahs will remain for as long as heaven and earth continue to exist. (Rav Se’adya Gaon, Sefer HaEmunot VehaDe’ot, Ma’amar 3)

“Good” refers to none other than Torah. For if people were to feel the sweetness and pleasantness of the Torah’s goodness, they would be crazed and wild in their pursuit of Torah. An entire world full of silver and gold would not be considered significant at all in their eyes, for the Torah includes all the goods in the world. (Rabbi Ĥayyim ben Atar, Or HaĤayyim, Ki Tavo 26:8)

By engaging in Torah study, one calls to God to come to him, so to speak, like a person who calls his friend to come to him, or a small child who calls his father to come and be together with him, and not leave him by himself, God forbid. (Rabbi Shneur Zalman of Liadi, Tanya, Likutei Amarim 37)

“Do not cast me off in old age” (Psalms 71:9). The words of Torah should not seem old to you. Rather, every day they should be new in your eyes. (Ba’al Shem Tov, Keter Shem Tov, Hosafot 66)

Every person has a special portion in the holy Torah that was allotted to him at Sinai. A subject matter that one has a strong desire to learn is the portion that he was allotted at Sinai. (Rabbi Menaĥem Mendel of Kotzk, Ohel Torah Nedarim)

Exerting effort in Torah study is not only a means to an end, so that one will thereby attain a high level of comprehension of Torah. It is an end in itself, as it is through exertion in Torah study that one merits the light of Torah and to cling to God. Without exertion in Torah study, even if one is intellectually gifted, he will not merit the hidden light that is concealed specifically for those who exert effort in Torah study. (Rabbi Shalom Noaĥ Berezovsky, Netivot Shalom, Part 1, Torah 4)

Just as a father buys a Humash for his young son and also hires him a tutor, so God, may He be blessed, both gave us the Torah and also, Himself, teaches each one of the children of Israel. This is [the meaning of] the blessing we recite: “The one who teaches Torah to the nation of Israel.” (Rabbi Yitzĥak Meir Alter, Ĥidushei HaRim)

It is the Torah of life – the Torah that gives instruction to each and every person about his day-to-day life. (Rabbi Yosef Yitzĥak Schneersohn)

“That you make us accustomed to Your Torah.” [This phrase in the prayers] is difficult. Didn’t the prophet rebuke Israel and say: “Their reverence of Me has become a commandment of men learned by rote” (Isaiah 29:13), which proves that habituation is not good?

Apparently it is only for mitzvot that it is not good, but for Torah study habituation is good. (Rabbi Menaĥem Mendel of Kotzk)

One who has not achieved the capacity to taste the World to Come, but his soul still longs and thirsts for God, pining for Him all day, and yet he does not quench his thirst in the waters of Torah available to him, is analogous to someone who is standing at a river but screams, “Water, [bring me] water to drink.” [Against this,] the prophet protests, “Ho, everyone thirsty, go to water” (Isaiah 55:1). (Rabbi Shneur Zalman of Liadi, Tanya, Likutei Amarim 40)

Further reading: The path to attaining excellence in Torah study is often not easy. Read anecdotes about great Torah figures and their path to Torah in A Concise Guide to the Sages, p. 330.