From where in the Torah does it mention shechita? And where does it mention it must be performed by a Jew? Yes this is a double question.
The answer to your second question is found at Why is meat only kosher if a Jew slaughters the animal?
The Rambam Hilchos Shechitah gives the mitzvos as
Introduction to Hilchos Shechitah
They contain 5 mitzvot: three positive commandments and two negative commandments. They are:
- To slaughter an animal and then to partake of it;
- Not to slaughter an animal and its offspring on the same day;
- To cover the blood of [slaughtered] beasts and fowl;
- Not to take a mother bird together with her young;
- To send away the mother when taking her together with her young.
These mitzvot are explained in the ensuing chapters
He then gives the source of the mitzvos as
It is a positive commandment1 for one who desires to partake of the meat of a domesticated animal, wild beast, or fowl to slaughter [it] and then partake of it,2 as Deuteronomy 12:21 states: "And you shall slaughter from your cattle and from your sheep." And with regard to a firstborn animal with a blemish,3 [ibid.:22] states: "As one would partake of a deer and a gazelle." From this, we learn that a wild beast is [governed by] the same [laws] as a domesticated animal with regard to ritual slaughter.
And with regard to a fowl, [Leviticus 17:13] states: "that will snare a beast or a fowl as prey... and shed its blood." This teaches that shedding the blood of a fowl is analogous to shedding the blood of a wild beast.4
1. Sefer HaMitzvot (positive commandment 146) and Sefer HaChinuch (mitzvah 451) include this among the 613 mitzvot of the Torah. As mentioned at the beginning of the Mishneh Torah, the Ra'avad differs and does not consider this a mitzvah.
2. The Rambam's wording echo his statements in Hilchot Berachot 11:2: "There are other mitzvot that are not obligations, but resemble voluntary activities, for example, the mitzvah of mezuzah.... A person is not obligated to dwell in a house that requires a mezuzah in order to fulfill this mitzvah." Similarly, in the instance at hand, a person is not obligated to slaughter. If, however, he desires to eat meat, he must fulfill this mitzvah.
3. Note the Kessef Mishneh who elaborates, explaining that although Rashi does not interpret the verse in the same manner the Rambam does, there is support for the Rambam's interpretation.
4. I.e., in both instances, ritual slaughter is required. The Kessef Mishneh notes that Chulin 27b derives this equivalence from another source and explains why the Rambam cites this verse instead.