In the Haftorah to Parashas Yisro, Isaiah 6:1 - 7:6, we read of Isaiah's prophetic vision where he visited Heaven. What Biblical and Talmudic personalities are said to have visited heaven and returned, either as a vision or in actuality?
Moshe Rabeinu went up to get the Torah and returned.
"שמות יט ג: "ומשה עלה
"שמות יט יד: "וירד משה
Rabbi Yishmael Kohain Gadol went up to check if there was a Gezaira on the Asara Haragai Malchus.
To add to the other answers, Rav Yosef the son of R' Yehoshua ben Levi once fainted and his soul temporarily departed from him (below is the gemara from Pesachim 50a paraphrased by Kollel Iyun Hadaf):
Question: "V'Hayah ba'Yom ha'Hu Lo Yihyeh Or Yekaros v'Kipa'on" - what are these?... R. Yehoshua ben Levi: "These are [rich] people who are Yekarim (important) in this world, but they are [not righteous, hence they will be] Kefuyim (light, unimportant) in the world to come," like Rav Yosef saw:
R. Yehoshua's son Yosef passed out; his spirit left, and returned. Yosef: "I saw an upside-down world - [esteemed] people on top here are below there, and vice-versa!" R. Yehoshua: "You saw reality! How did Chachamim look?" Yosef: "They are as [important as] they are here. They were saying that happy is the one who remembers his learning when he comes to the world to come. They also said that no one else has a portion like those [who sanctified Hashem's name when] killed by the king."
To which people did he refer? He did not mean R. Akiva and the other [nine] Chachamim killed by the king - even if they were not killed by the king, no one else would have a portion like them! Rather, it referred to those killed in Lud. (Rashi - the king made a false accusation as a pretext to kill all the Yisraelim; two brothers confessed to it to save everyone else; Ran - there was a decree to drink from colored glass Kelim, unlike the practice of Yisrael - Yisraelim refused to change and submitted to be killed.)
The Gemara elsewhere relates that R' Yehoshua ben Levi himself made it into Gan Eden alive (below is the gemara from Kesuvos 77b paraphrased by Kollel Iyun Hadaf):
When R. Yehoshua ben Levi was about to die, the Satan (angel of death) was told to accede to his desires. The Satan appeared to him. R. Yehoshua ben Levi asked to see where his place will be (in Gan Eden). The Satan consented. R. Yehoshua ben Levi asked for the Satan's knife, lest it scare him on the way; he gave it to him. When they got there, the Satan lifted him up and showed him. R. Yehoshua ben Levi jumped to the other side, held onto his garment and swore that he will not leave. Hash-m said: "If he ever annulled an oath, he will go back; if not, not." (He had not.) The Satan asked him to return his knife. R. Yehoshua ben Levi refused. A voice from Heaven: "Return it, for it is needed for the world." (R. Yehoshua ben Levi returned it.)
Eliyahu said: "Make room for the son of Levi! Make room for the son of Levi!" R. Yehoshua ben Levi found R. Shimon bar Yochai sitting on 13 gold chairs. R. Shimon bar Yochai: "Are you the son of Levi?" R. Yehoshua ben Levi: "Yes." R. Shimon bar Yochai: "Was a rainbow seen in your day?" R. Yehoshua ben Levi: "Yes." R. Shimon bar Yochai: "If so, you are not worthy of this proclamation." (The rainbow is Hashem's sign not to destroy the world. Your merit did not suffice to protect the world!) Really, the rainbow was not seen in his day. R. Yehoshua ben Levi did not want to take credit for this.
Avraham Avinu. See Rashi Bereshit 15:5
וַיּוֹצֵא אֹתוֹ הַחוּצָה וַיֹּאמֶר הַבֶּט נָא הַשָּׁמַיְמָה וּסְפֹר הַכּוֹכָבִים אִם תּוּכַל לִסְפֹּר אֹתָם וַיֹּאמֶר לוֹ כֹּה יִהְיֶה זַרְעֶךָ:
ויוצא אתו החוצה: .... דבר אחר הוציאו מחללו של עולם והגביהו למעלה מן הכוכבים, וזהו לשון הבטה מלמעלה למטה:
And He took him outside: .... Another explanation: He took him out of the terrestrial sphere and lifted him above the stars. This explains the expression of הַבָּטָה, looking down from above (Gen. Rabbah 44:12).
Here is a well known story about 4 who went to heaven http://www.chabad.org/kabbalah/article_cdo/aid/380344/jewish/Four-Who-Entered-Paradise.htm
this is the opening to that webpage (there is a lot more about them and even a fictionalized account in a book if you want to delve):
The Talmud (Chagiga 14b, Zohar I, 26b and Tikunei Zohar, Tikun 40) reports the following incident regarding four Mishnaic Sages:
The Rabbis taught: Four [Sages] entered the Pardes [literally "the orchard."]. Rashi explains that they ascended to heaven by utilizing the [Divine] Name [i.e., they achieved a spiritual elevation through intense meditation on G-d's Name] (Tosafot, ad loc). They were Ben Azzai, Ben Zoma, Acher [Elisha ben Avuya, called Acher-- the other one -- because of what happened to him after he entered the Pardes] and Rabbi Akiva.
Although the question asks about Biblical and talmudic personalities, I think it's worthwhile to note that tzaddikim way after this period visited heaven.
Baal Shem Tov for instance.
The Baal Shem Tov used Kabbalah to experience an elevation of the soul to the heavenly realms. It was here that he met with Mashiach. The Baal Shem Tov asked, “When will you arrive?” Mashiach replied, “When your teachings will be widely disseminated.”
The above (abridged) account is recorded in a letter from the Baal Shem Tov (brought in the sefer Keter Shem Tov in the beginning)
ועליתי מדריגה אחר מדריגה, עד שנכנסתי להיכל משיח ששם לומד משיח תורה עם כל התנאים והצדיקים וגם עם שבעה רועים, ושם ראיתי שמחה גדולה עד מאד, ואיני יודע לשמחה זו מה הוא עושה, והייתי סובר שהשמחה הזו ח"ו על פטירתי מהעוה"ז, והודיעו לי אח"כ שאיני נפטר עדיין כי הנאה להם למעלה כשאני מייחד יחודים למטה ע"י <תורתם> [תורותיהם] הקדושה, אבל מהות השמחה איני יודע עד היום הזה.