A person who, for example, steals, is obligated to apologise and offer to return what was stolen or compensate the owner. Is an adult who transgressed as a minor also obligated to do teshuva?
(The TL;DR: Teshuvah for the sin, no. Teshuvah for any trace of the personality still around that was able to do that sin, good idea. Closed with personal story.)
The Rambam discusses two kinds of teshuvah. Hilkhos Teshuvah opens (ch. 1-2) with a discussion of abandoning one's bad actions. Teshuvah for what we did, breaking bad habits, and taking steps to do better in the future.
Then there is the teshuvah the Rambam introduces in 7:3:
אַל תֹּאמַר שֶׁאֵין תְּשׁוּבָה אֶלָּא מַעֲבִירוֹת שֶׁיֵּשׁ בָּהֶן מַעֲשֶׂה כְּגוֹן זְנוּת וְגֵזֶל וּגְנֵבָה. אֶלָּא כְּשֵׁם שֶׁצָּרִיךְ אָדָם לָשׁוּב מֵאֵלּוּ כָּךְ הוּא צָרִיךְ לְחַפֵּשׂ בְּדֵעוֹת רָעוֹת שֶׁיֵּשׁ לוֹ וְלָשׁוּב מִן הַכַּעַס וּמִן הָאֵיבָה וּמִן הַקִּנְאָה וּמִן הַהִתּוּל וּמֵרְדִיפַת הַמָּמוֹן וְהַכָּבוֹד וּמֵרְדִיפַת הַמַּאֲכָלוֹת וְכַיּוֹצֵא בָּהֶן מִן הַכּל צָרִיךְ לַחֲזֹר בִּתְשׁוּבָה. וְאֵלּוּ הָעֲוֹנוֹת קָשִׁים מֵאוֹתָן שֶׁיֵּשׁ בָּהֶן מַעֲשֶׂה שֶׁבִּזְמַן שֶׁאָדָם נִשְׁקָע בְּאֵלּוּ קָשֶׁה הוּא לִפְרשׁ מֵהֶם. וְכֵן הוּא אוֹמֵר (ישעיה נה ז) "יַעֲזֹב רָשָׁע" וְגוֹ':
A person should not think that repentance is only necessary for those sins that involve deed such as promiscuity, robbery, or theft. Rather, just as a person is obligated to repent from these, similarly, he must search after the evil character traits he has. He must repent from anger, hatred, envy, frivolity, the pursuit of money and honor, the pursuit of gluttony, and the like. He must repent for all [of the above].
These sins are more difficult than those that involve deed. If a person is attached to these, it is more difficult for him to separate himself. In this context, [Isaiah 55:7] exhorts: "May the wicked abandon his path and the crooked man, his designs."
As Salmononius2 wrote in his answer and robev commented, the Rama (OC 343:1) says one is not obligated to do teshuvah for violating kibud av va'eim (parental respect) as a child. And the Qitzur (165:6) gives the logical generalization of applying this to other sins.
However, the Rama doesn't stop there. This is what he writes:
וקטן שהכה לאביו או עבר שאר עבירות בקטנותו אע"פ שא"צ תשובה כשהגדיל מ"מ טוב לו שיקבל על עצמו איזה דבר לתשובה ולכפרה אע"פ שעבר קודם שנעשה בר עונשין (פסקי מהרא"י סי' ס"ב):
If a child struck his father or transgressed otherwise in his youth, even though he does not need to repent when he matures, it is still good for him to take upon himself some element of teshuvah and atonement, even though his sin was at age when he was not eligible for punishment (Pisqei Mahara"i #62).
I would suggest this is because of the existence of two types of teshuvah. Deeds done as children aren't true sins, and don't require teshuvah. But those deeds did reinforce negative elements of one's character. If one feels those flaws are still around, it would be appropriate to do teshuvah in the sense of repairing one's middos.
And now let me add a personal note. I changed schools for 5th grade. In the first school I attended, the bullying started in 2nd grade, but by the end of fourth grade... Two kids tripped me on the staircase, and I fell down a flight of concrete steps. So, it was time to go. That was back in 1975.
On Erev Yom Kippur, in 2008, I got a call from someone I hadn't heard from since then. No, not one of the boys who tripped me, but still, someone who had made my life miserable. He asked for my mechilah (forgiveness).
In truth, it didn't cross my mind that the 43 year old who called me would be the same person as the 7 to 9 year old who bullied me. So, granting it was easy.
But his willingness to right an old wrong, hunt down where I am living now and how to reach me, and admit to what he was like... I mean, most of us would be happy to be off the hook, no?
It was a lesson in teshuvah I took with me.
From the Kitzur Shulchan Aruch 165:6, it is implied that there is no actual obligation for an adult to do Teshuva for the transgressions they did as a child, but it is recommended for the adult to "accept upon themselves some sort of repentance".
Here is the quote (emphasis mine):
קָטֹן שֶׁגָּנַב אֵיזֶה דָבָר, אִם הוּא בְעָיִן, מְחֻיָבִין לְהַחֲזִירוֹ. וְאִם אֵינוֹ בְעָיִן, פָּטוּר מִדִינֵי אָדָם אַף לְאַחַר שֶׁיִגְדַּל. אַךְ לָצֵאת יְדֵי שָׁמַיִם, חַיָב לְשַׁלֵם כְּשֶׁיִגְדָּל. וְכֵן אִם עָשָׂה שְׁאָר עֲבֵרוֹת בְקַטְנוּתוֹ כְּשֶׁהוּא בַּר הֲבָנָה, טוֹב שֶׁיְקַבֵּל עָלָיו אֵיזֶה דָבָר לִתְשׁוּבָה. וְעַל זֶה נֶאֱמַר, גַּם בְּלֹא דַעַת נֶפֶשׁ לֹא טוֹב.
If a child steals something, if it is still intact, you are obligated to return it; but if it is no longer intact, he is legally exempt from making restitution even after he becomes of age. But in order to fulfill his obligation before the judgement in Heaven, he must make restitution when he becomes of age. Similarly, if he committed other sins in his youth [before Bar Mitzvah], it is advisable that he accept upon himself some sort of repentance when he reaches an age of understanding. Concerning this it is said, "For the soul to be without knowledge is not good."13