The Torah portion is Shemini, Leviticus 9:1-11:47. In synagogues using the Triennial system, 5782 is year 3 and the Torah reading is Leviticus 11:1-11:47.
This parshah speaks of the offerings on the eighth day, that is, the day after the seven days of consecration of Aaron and the other priests. But it also speaks of the unauthorized offerings of Aaron's sons, Nadav and Abihu, who offered "strange" fire and were themselves consumed by fire. Aaron and his surviving sons were told not to observe normal mourning rituals for Nadav and Abihu. It goes on to speak of offerings that are normally eaten by the priests. It lays out the rules of kosher meat, fish, birds and other species, and other rules about kosher food.There is a special maftir portion (the final piece of the Torah reading) for Shabbat Parah: Numbers 19:1-22. This passage speaks of the offering of the Red Heifer (Parah Adumah), which purifies people from contact with the dead. It is a reminder to purify for Pesach (Passover), which is coming very soon!
The haftarah is for Shabbat Parah, Ezekiel 36:16-38. Like the maftir portion, this haftarah speaks of purification.
In synagogues using the Sephardic tradition, the Haftarah is Ezekiel 36:16-36.