Women's Torah education in Judaism has been the subject of debate from Mishnaic times right down to the present day.
As the Mishnaic and Talmudic literature are primarily legal texts, they are more concerned with what the law is regarding women's education than discussing history.
There are a couple of primary sources that discuss the laws of women's education. The Mishnah (Kiddushin 1:7) discusses the general principles by which we determine whether a commandment applies to men or women (or both):
כל מצות הבן על האב אנשים חייבין ונשים פטורות וכל מצות האב על הבן אחד
אנשים ואחד נשים חייבין וכל מצות עשה שהזמן גרמא אנשים חייבין ונשים
פטורות וכל מצות עשה שלא הזמן גרמא אחד האנשים ואחד הנשים חייבין וכל
מצות לא תעשה בין שהזמן גרמא בין שלא הזמן גרמא אחד האנשים ואחד הנשים
חייבין חוץ מבל תקיף ובל תשחית ובל תטמא למתים
MISHNAH. ALL OBLIGATIONS OF THE SON UPON THE FATHER, MEN ARE BOUND,
BUT WOMEN ARE EXEMPT. BUT ALL OBLIGATIONS OF THE FATHER UPON THE SON,
BOTH MEN AND WOMEN ARE BOUND. ALL AFFIRMATIVE PRECEPTS LIMITED TO
TIME, MEN ARE LIABLE AND WOMEN ARE EXEMPT. BUT ALL AFFIRMATIVE
PRECEPTS NOT LIMITED TO TIME ARE BINDING UPON BOTH MEN AND WOMEN. AND
ALL NEGATIVE PRECEPTS, WHETHER LIMITED TO TIME OR NOT LIMITED TO TIME,
ARE BINDING UPON BOTH MEN AND WOMEN; EXCEPTING, YE SHALL NOT ROUND
[THE CORNERS OF YOUR HEADS], NEITHER SHALT THOU MAR [THE CORNER OF THY
BEARD], AND, HE SHALL NOT DEFILE HIMSELF TO THE DEAD. (Soncino
translation; capitals in original)
The Talmudic commentary to this Mishnah (Kiddushin 29a) cites a Beraisa which states:
האב חייב בבנו למולו ולפדותו וללמדו תורה ולהשיאו אשה וללמדו אומנות וי"א
אף להשיטו במים רבי יהודה אומר כל שאינו מלמד את בנו אומנות מלמדו ליסטות
The father is bound in respect of his son, to circumcise, redeem,
teach him Torah, take a wife for him, and teach him a craft. Some say,
to teach him to swim too, R. Judah said: He who does not teach his son
a craft, teaches him brigandage (Soncino translation)
The Beraisa only makes note of an obligation to teach one's son, but says nothing about teaching one's daughter.
The Talmud on the next page (Kiddushin 29b) explains the Scriptural source, and demonstrates that women are not obligated to learn, teach, or be taught:
ללמדו תורה מנלן דכתיב ולמדתם אותם את בניכם והיכא דלא אגמריה אבוה
מיחייב איהו למיגמר נפשיה דכתיב ולמדתם איהי מנלן דלא מיחייבא דכתיב
ולימדתם ולמדתם כל שמצווה ללמוד מצווה ללמד וכל שאינו מצווה ללמוד אינו
מצווה ללמד ואיהי מנלן דלא מיחייבה למילף נפשה דכתיב ולימדתם ולמדתם כל
שאחרים מצווין ללמדו מצווה ללמד את עצמו וכל שאין אחרים מצווין ללמדו אין
מצווה ללמד את עצמו ומנין שאין אחרים מצווין ללמדה דאמר קרא ולמדתם אותם
את בניכם ולא בנותיכם
‘To teach him Torah.’ How do we know it? — Because it is written. And
ye shall teach them your sons. And if his father did not teach him, he
must teach himself, for it is written, and ye shall study. How do we
know that she [the mother] has no duty [to teach her children]? —
Because it is written, we-limaddetem [and ye shall teach], [which also
reads] u-lemadetem [and ye shall study]: [hence] whoever is commanded
to study, is commanded to teach; whoever is not commanded to study, is
not commanded to teach. And how do we know that she is not bound to
teach herself? — Because it is written, we-limaddetem [and ye shall
teach] — u-lema — detem [and ye shall learn]: the one whom others are
commanded to teach is commanded to teach oneself; and the one whom
others are not commanded to teach, is not commanded to teach oneself.
How then do we know that others are not commanded to teach her? —
Because it is written: ‘And ye shall teach them your sons’ — but not
your daughters. (Soncino translation)
The above establishes that on a Biblical level, women are exempt from education. However, the Sages debated whether women should be taught anyway because of other factors. A Mishnah (Sotah 3:4) states:
אם יש לה זכות היתה תולה לה יש זכות תולה שנה אחת יש זכות תולה ב' שנים
יש זכות תולה ג' שנים מכאן אומר בן עזאי חייב אדם ללמד את בתו תורה שאם
תשתה תדע שהזכות תולה לה ר"א אומר כל המלמד בתו תורה לומדה תפלות
IF SHE POSSESSED A MERIT, IT [CAUSES THE WATER] TO SUSPEND ITS EFFECT
UPON HER. SOME MERIT SUSPENDS THE EFFECT FOR ONE YEAR, ANOTHER FOR TWO
YEARS, AND ANOTHER FOR THREE YEARS. HENCE DECLARED BEN AZZAI, A MAN IS
UNDER THE OBLIGATION TO TEACH HIS DAUGHTER TORAH, SO THAT IF SHE HAS
TO DRINK [THE WATER OF BITTERNESS], SHE MAY KNOW THAT THE MERIT
SUSPENDS ITS EFFECT. R. ELIEZER SAYS: WHOEVER TEACHES HIS DAUGHTER
TORAH TEACHES HER OBSCENITY. (Soncino translation; capitals in
original)
In this source we find on the one hand that some of the Sages recognized that due to certain exigencies teaching Torah to women should be a necessity, while the other Sages felt that teaching Torah to women would lead to disastrous consequences.
This latter sentiment is echoed in the Jerusalem Talmud (Sotah 3:4) in which some of the Sages state that women's wisdom is limited to the spindle, and the Torah should be burned rather than given over to women:
מטרונה שאלה את רבי לעזר מפני מה חט אחת במעשה העגל והן מתים בה שלש
מיתות אמר לה אין חכמתה של אשה אלא בפילכה דכתיב וכל אשה חכמת לב בידיה
טוו אמר לו הורקנוס בנו בשביל שלא להשיבה דבר אחד מן התורה איבדת ממני
שלש מאות כור מעשר בכל שנה אמר ליה ישרפו דברי תורה ואל ימסרו לנשים
וכשיצתה אמרו לו תלמידיו רבי לזו דחיתה לנו מה אתה משיב
These sources seem to indicate that teaching women was encouraged by some and frowned upon by others, which may mean that some women (during certain periods) were taught, while others were not.
Another Mishnaic source about women's education appears in Nedarim (4:3). The Mishnah states:
ותורם את תרומתו ומעשרותיו לדעתו ומקריב עליו קיני זבין קיני זבות קיני
יולדות חטאות ואשמות ומלמדו מדרש הלכות ואגדות אבל לא ילמדנו מקרא אבל
מלמד הוא את בניו ואת בנותיו מקרא
MISHNAH. AND HE MAY SEPARATE HIS TERUMAH AND HIS TITHES WITH HIS
CONSENT. HE MAY OFFER UP FOR HIM THE BIRD SACRIFICES OF ZABIM AND
ZABOTH AND THE BIRD SACRIFICES OF WOMEN AFTER CHILDBIRTH,
SIN-OFFERINGS AND GUILT-OFFERINGS. HE MAY TEACH HIM MIDRASH, HALACHOTH
AND AGGADOTH, BUT NOT SCRIPTURE. YET HE MAY TEACH SCRIPTURE TO HIS
SONS AND DAUGHTERS. (Soncino translation; capitals in original)
This indicates that it was allowed, and perhaps even normal, to teach girls (at least Scripture if nothing else). However, the text of the Mishnah in Maimonides's version does not contain the words "ואת בנותיו" "and his daughters". The words are similarly absent in this early manuscript.
Even if the words were not part of the Mishnah, though, the subsequent Talmudic analysis of this Mishnah assumes that girls were included.
Finally, there is a Talmudic source that explicitly mentions the education of a specific woman, and even holds her as a standard for male scholars. In Pesachim (62b) we find:
ר' שמלאי אתא לקמיה דרבי יוחנן א"ל ניתני לי מר ספר יוחסין א"ל מהיכן את
א"ל מלוד והיכן מותבך בנהרדעא א"ל אין נידונין לא ללודים ולא לנהרדעים
וכל שכן דאת מלוד ומותבך בנהרדעא כפייה וארצי א"ל ניתנייה בג' ירחי שקל
קלא פתק ביה א"ל ומה ברוריה דביתהו דר"מ ברתיה דר"ח בן תרדיון דתניא תלת
מאה שמעתתא ביומא מג' מאה רבוותא ואפ"ה לא יצתה ידי חובתה בתלת שנין ואת
אמרת בתלתא ירחי
R. Simlai came before R. Johanan [and] requested him, Let the Master
teach me the Book of Genealogies. Said he to him, Whence are you? — He
replied, From Lod. And where is your dwelling? In Nehardea. Said he to
him, We do not discuss it either with the Lodians or with the
Nehardeans, and how much more so with you, who are from Lod and live
in Nehardea! But he urged him, and he consented, Let us learn it in
three months, he proposed. [Thereupon] he took a clod and threw it at
him, saying, If Beruriah, wife of R. Meir [and] daughter of R. Hanina
b. Teradion, who studied three hundred laws from three hundred
teachers in [one] day, could nevertheless not do her duty in three
years, yet you propose [to do it] in three months! (Soncino
translation)
While it is certainly possible, even likely, that Beruriah was the exception rather than the rule, it does at least indicate that unique women could be given an education.
The debate about women's education is still ongoing, with some streams of Judaism offering girls almost no formal (textual) Torah education, others teaching girls the same way they teach boys, and many gradations in between. As the question here seems to be about education during the Talmudic Era, the post-Talmudic sources would seem to be largely irrelevant.
To summarize:
There is not a tremendous amount of historical material in the Talmudic Era literature. We do see that the consensus was that women are exempt from Torah education, with a dispute as to whether they should be taught anyway. It appears that some amount of Torah education was considered lawful, and perhaps even normal, and individual women may have been heavily educated.
I subsequently found that a footnote in the Soncino translation of one of the above cited sources (Nedarim 35b note 4) makes almost all the points in my answer with the same citations:
From this we see that it was usual to teach the Bible to girls, in
spite of the Talmudic deduction that daughters need not be educated
(Kid. 30a). The opposition of R. Eliezer to teaching Torah to one's
daughter (Sot. 20a: He who teaches his daughter Torah is as though he
taught her lewdness) was probably directed against the teaching of the
Oral Law, and the higher branches of study. [V. Maim. Yad. Talmud
Torah, I, 13.] Yet even in respect of this, his view was not
universally accepted, and Ben ‘Azzai (a.l.) regarded it as a positive
duty to teach Torah to one's daughters. The context shows that the
reference is to the higher knowledge of Biblical law. In point of
fact, there were learned women in Talmudic times e.g., Beruriah, wife
of R. Meir (Pes. 62b).