Parsha Re-ei, like the rest of the book of Devarim, has Moshe Rabeinu addressing the Jewish nation just before he is to pass away, and just before the Jews cross the Jordan River and enter the land of Israel.
The first aliyah begins with the Jews being told of a blessing if they obey G-d's commandments, and a curse if they don't. Their instructions are that they should destroy all the idols and altars found in the land of Israel. Then they are told that G-d will, in the future, designate a specific place for sacrifices and other avodah (service to G-d) that they presently do in the Mishkan, and of course, that place will turn out to be Yirushalayim (Jerusalem).
The second aliyah again refers to the place that G-d will in the future designate as the one place to offer sacrifices, which will be Yirushalayim.
The second aliyah continues by discussing the slaughter of meat for ordinary consumption. In Chapter 12, pussok 21, G-d says that we may slaughter animals in the manner that was described (previously), and this is a place where the written Torah refers to the Oral Torah which was also given to Moshe on Mt. Sinai. This aliyah also says that we should not eat the blood of animals, and that we should observe and hear all the commandments. Rashi says this refers to the study of the Oral Law. The Torah says that the reward for both of these is that it should go well with us and for our children after us.
In the third aliyah we are told to not follow after the ways of the idol worshipers that are to be removed from Israel before we get there. We are not even allowed to inquire about their way of serving their idols. Here we are also told to not believe a false prophet, even if he correctly predicts a sign or a wonder. Furthermore, if someone close to you tries to draw you to idol worship, you shouldn't listen, and such an act is punishable by death.
The fourth aliyah contains several commandments that distinguish the Jewish nation from the other nations. As chosen people we are not allowed to mutilate our bodies for the dead as some nations are accustomed to do. Furthermore, we eat only animals with a wholly cloven hoof, and that chew their cud. Fish must have fins and scales. Birds that we may not eat are listed here. Meat must be properly slaughtered. Meat must not be cooked with milk.
The fifth aliyah discusses the requirement of the second tithe of the produce of our crops (the first tithe is the one given to the Levites). We are to bring the second tithe to Yirushalayim and eat it there and rejoice there. Provision is made here for people who live far away from Yirushalayim and who have lots of goods to carry all that way. They can sell it locally (where they live), and take the money to Yirushalayim instead, and buy good food and drink with the money and eat and rejoice in Yirushalayim.
The sixth aliyah contains a law of the Shmittah year: Every seven years creditors are to forgive outstanding loans to Jews. This aliyah says that if we do this, there will be no poor among us and that G-d will bless us as follows: we shall lend to many nations, but none shall need to lend to us, and we shall rule over many nations, but none shall rule over us.
Also in this aliyah are some laws of tzedakah. Rashi derives from these laws that tzedakah to the poor of one's family has preference over other poor of one's city, and the poor of one's city has preference over other cities. The law of the Jewish slave is given here.
The seventh aliyah contains the law of the first-born of our cattle and flocks. Providing the animal has no blemish, we must do no work with it and we should bring it to Yirushalayim and sacrifice it and eat it there.
The three holidays of Pesach and Shavous and Sukkos are described here.