So you're curious about peyos, but you don't come from a custom that wears them.
Firstly, I want to just gently offer the concept that we generally follow the customs of our congregation or rabbi. So if it's not part of your tradition, consider why you want to take this on. What are you committing to by engaging in this mitzvah? What potential risks or losses might you experience? When taking on a new mitzvah, especially one that's very visible in public, you should be careful about whether you really want to commit to it. But if, after all is said and done, you still are interested in wearing peyos, you should know the rules and the norms around them.
You shall not round off the side-growth on your head, or destroy the side-growth of your beard.
They shall not shave smooth any part of their heads, or cut the side-growth of their beards, or make gashes in their flesh.
Koren translation of Leviticus 21:5:
They shall not make baldness on their head, neither shall they shave off the corner of their beard, nor make any cuttings in their flesh.
Next, let's go to the Talmud. The Rabbis are trying to figure out if women are prohibited from cutting their hair in certain ways, or from shaving any facial hair they may grow.
The mishna teaches that women are obligated in all prohibitions except for the prohibitions of: Do not round the corners of one’s head, and: Do not destroy the corners of your beard, and: Do not contract ritual impurity from a corpse. The Gemara asks: Granted, a woman of priestly lineage is not obligated in the mitzva of: Do not contract ritual impurity from a corpse, as it is written: “Speak to the priests, the sons of Aaron, and say to them: None shall become impure for the dead among his people” (Leviticus 21:1). This verse teaches that the prohibition applies to the sons of Aaron, but not the daughters of Aaron. But from where do we derive the prohibitions of: Do not round the corners of one’s head, and: Do not destroy the corners of your beard? The Gemara answers that this is as it is written: “You shall not round the corners of your head and you shall not destroy the corners of your beard” (Leviticus 19:27). The juxtaposition of the two prohibitions teaches that anyone who is included in the prohibition against destroying the beard is included in the prohibition against rounding the head. And since these women are not included in the prohibition against destroying, they are also not included in the prohibition against rounding the head. The Gemara asks: And from where do we derive that women are not obligated in the prohibition against destroying the corners of one’s beard? The Gemara answers: If you wish, propose a logical reason, as ordinarily women do not have a beard. And if you wish, cite a verse that teaches this point, as the verse states: “You shall not round the corners of your head [roshekhem] and you shall not destroy the corners of your beard [zekanekha]” (Leviticus 19:27). The Gemara explains: From the fact that the verse changed its language, as the term “your head [roshekhem]” is in the plural while “your beard [zekanekha]” is in the singular, it can be inferred that if so, if the prohibition against destroying one’s beard applied to everyone, let the Merciful One write: And you shall not destroy the corners of your beards [zekanekhem], in the plural, so that the end of the verse parallels the beginning. What is indicated by the fact that the verse states: “And you shall not destroy the corners of your beard [zekanekha],” in the singular? This serves to teach: Your beard is included, but not your wife’s beard.
(א) אֵין מְגַלְּחִין פַּאֲתֵי הָרֹאשׁ כְּמוֹ שֶׁהָיוּ עוֹשִׂין עוֹבְדֵי כּוֹכָבִים שֶׁנֶּאֱמַר (ויקרא יט כז) "לֹא תַקִּפוּ פְּאַת רֹאשְׁכֶם". וְחַיָּב עַל כָּל פֵּאָה וּפֵאָה. לְפִיכָךְ הַמְגַלֵּחַ שְׁנֵי צְדָעָיו אֲפִלּוּ בְּבַת אַחַת וְהַתְרָאָה אַחַת לוֹקֶה שְׁתַּיִם. אֶחָד הַמְגַלֵּחַ הַפֵּאוֹת בִּלְבַד וּמֵנִיחַ שֵׂעָר כָּל הָרֹאשׁ וְאֶחָד הַמְגַלֵּחַ כָּל הָרֹאשׁ כְּאֶחָד לוֹקֶה הוֹאִיל וְגִלַּח הַפֵּאוֹת. בַּמֶּה דְּבָרִים אֲמוּרִים בְּאִישׁ הַמְגַלֵּחַ אֲבָל אִישׁ הַמִּתְגַּלֵּחַ אֵינוֹ לוֹקֶה אֶלָּא אִם כֵּן סִיֵּעַ לַמְגַלֵּחַ. וְהַמְגַלֵּחַ אֶת הַקָּטָן לוֹקֶה:
(1) We may not shave [but may cut with scissors] the corners of our heads as the idolaters and their priests do, [Sefer HaChinuch (Mitzvah 251) explains that this is a particularly severe prohibition, since its violation involves making a sign for idolatry on our own bodies] as [Leviticus 19:27] states: "Do not cut off the corners of your heads."
One is liable for each corner. Therefore, a person who shaves both his temples
[[This prohibition applies equally to] one who shaves off only the corners of his head and leaves the remainder of his hair - As mentioned in Chapter 11, Halachah 1, this style of cutting hair is referred to as a blorit and was practiced by the gentiles] - even if he were to do so simultaneously and had received only a single warning – is [liable for] two measures of lashes.
[This prohibition applies equally to] one who shaves off only the corners of his head and leaves the remainder of his hair, and to one who shaves his entire head at once. Since he has shaved the corners, he is [liable for] lashes. [From this, we see that the mitzvah is not dependent on the rationale of "not appearing like a gentile."]
To whom does the above apply? To the person who shaves. The person [whose head] is shaven is not lashed [The Ra'avad maintains that this person is not punished because he did not perform a deed. He is, however, considered to have transgressed the Torah's prohibition. The Kessef Mishneh disagrees and maintains that since he did not perform the deed of shaving, he is not considered to have violated the prohibition at all. This applies even when he specifically ordered the person who shaved him to do so. The Ra'avad's opinion is, however, supported by the Lechem Mishneh and other authorities] unless he assists the one who is shaving him. One who shaves [the corners of] a child's [head] should be [liable for] lashes. [A child would not be held responsible if he shaved himself, because a child is not held liable for the violation of any of the Torah's prohibitions until he reaches majority. Nevertheless, an adult is responsible for shaving the child's head (Nazir 57b).
This point is not, however, accepted by all authorities. The Beit Yosef (Yoreh De'ah, ibid.) mentions other opinions which do not hold a person liable for shaving a child's head.]
ע”ד הפשט: טעם האסור כדי שלא יבטל הסימן שהקב”ה רשם במין הזכר כדי להבדילו ממין הנקבה, כי העושה כן הפך השם הוא עושה, כמי שזורע כלאים, ובכל מה שעשה במעשה בראשית כתב בהם “למינהו
A very different approach comes from R. Bachya ben Asher (1255-1340), who suggests that beard and side-locks are one of the main ways of differentiating between men and women.Thus, obliterating these markers would create a gap in the gender divide, something the Torah is wary of in other places as well (see esp. Deuteronomy 22:5, where men and women are forbidden to cross dress.)
From The Prohibition of Shaving in the Torah and Halacha Dr. Rabbi Zev Farber:
According to the halacha, as recorded above, it is forbidden for a man to shave off the hair of his temples or to shave off the corners of his beard.[2] Both of these rules apply only to shaving with a razor. Using a depilating cream, a waxing process, or just pulling the hair out by hand would not be forbidden. Trimming with scissors and most forms of electric shavers (where the razor does not touch the face) are not forbidden either.