In the first aliyah, we find the Jewish people a mere three days away from their entry to the promised land, and as Rashi says, the Jews went to Moshe and asked if they could send spies to check out the land they were going to.
Moshe tells them there is no need for this because G-d promised them a good land. But G-d gives them free will to err this way, and in this aliyah G-d gives permission for the Jews to send spies to spy out the land. Rashi points out that this section is right next to the section in the previous parsha about Miriam's loshen hora and her punishment, to give the spies an opportunity to learn from her.
Due to their actions in this parsha the Jewish people will have to wander in the desert for 40 years before reaching the land of Israel.
The second aliyah begins the drama of the story of the spies. They enter the land, and Rashi says that Calev, one of the spies, goes alone to Chevron where the patriarchs are buried, to pray that his companions shouldn't persuade him to do wrong.
The spies cut a huge cluster of grapes and carry it on a pole (this is a symbol of the Israeli tourist organization). The spies come back after 40 days. Rashi says this trip should have taken 80 days, but because G-d was going to keep the Jews in the desert one year for every day they were away, G-d shortened their trip.
The spies come back to the Jews in the desert and report that it is indeed a land flowing with milk and honey, but that it is well fortified and impossible to conquer. Calev speaks up and points out that Moshe split the sea, and got the manna and the quail, and that at his word anything is possible. He finished by saying that there is nothing to worry about. The Jewish people conclude that they would rather go back to Egypt. They spend that night wailing. This wailing takes place the night of the 9th of Av, and G-d says that if the people choose to wail for no reason on the 9th of Av, then He will give them something to wail about. As a result, both the first and second Bais Hamikdash's were destroyed, and other afflictions for Jewish people took place on that date, including WW-II, which began on that day too.
In the third aliyah, Calev and Yehoshua, who both began speaking near the end of the second aliyah, continue speaking in this aliyah and beg the people to trust in G-d to bring them to the land flowing with milk and honey. Furthermore, they beg the people to not rebel against G-d and Moshe. But the Jewish people reject their plea and decide to stone them.
G-d then appears in a cloud and asks Moshe how long will these people despise G-d and not believe in all the miracles G-d has done for them. G-d says he will destroy the people and raise a new nation mightier than them. But Moshe points out that it won't look good to the Egyptians that G-d took the Jewish people out of Egypt only to kill them in the desert.
Moshe begs for G-d's mercy. G-d then decides that this generation will wander in the desert for forty years (one year for each day the spies traveled). All this generation, except for Calev and Yehoshua, will die in the desert and not enter into the Promised Land. That is their punishment for believing the loshen hora of the spies and not trusting in G-d.
In the fourth aliyah G-d tells how the punishment will take place. All people that were numbered by counting shekels, which were the men between the ages of 20 and 60, are to die in the desert; while all the women, and the Leviim, and Calev and Yehoshua, and the children, will not die, and will wander in the desert for 40 years, and then go up into the promised land. The ten spies who spoke evil about Eretz Yisroel die immediately of a terrible plague.
A group of people, who are now supposed to wander and die in the desert, regret their sin and rise up early the next morning and attempt to go straight the three days journey to Israel. Moshe warns them that it won't work because G-d is not with them in what they are doing. They attempt to go anyway, but the nation of Amalak attacks and kills them immediately.
At the end of the previous aliyah, and continuing in this, the fifth aliyah, G-d gives the laws of drink offerings and flour offerings accompanying certain sacrifices. It may be asked: what are these doing here right after the story of the spies and the punishment decreed on the Jews because of it?
It says in the Midrash that after these events the Jews become depressed and begin mourning their mistake and their destiny to wander and die in the desert. They reason further that perhaps even their children will make a mistake too and our people will never reach the land. Chassidus teaches that depression may not be a sin, but it is worse than any sin, because it can lead to all the other sins. So G-d asks Moshe to cheer up the Jews and told him to do this by teaching these mitzvahs which apply mainly in Eretz Yisroel. In this way, the Jews will be confident that their children will reach Israel to do these mitzvahs.
The sixth aliyah continues giving additional mitzvot. Here the Torah gives the mitzvah of taking challah from dough. This is one of three main mitzvot of woman, the other two being lighting Shabbos candles and Mikvah, the G-d given gift to the Jewish people, enhancing the spiritual and physical health of the marriage and the family. After this is the law of a community inadvertently committing an act of idolatry because of their leadership. The Torah says here that an act of idolatry is equivalent to violating all the commandments.
The seventh aliyah gives the law of an individual committing an act of idolatry inadvertently, and also if he does it, G-d forbid, on purpose. Then an incident is described where a man is found gathering sticks on Shabbos. Rashi says that this happened on the second Shabbos after the giving of the Torah. Therefore the Jewish people properly observed only one Shabbos until today. It says in the Talmud that if the Jewish people observed even two Shabbos's properly (which means just one more), we would immediately be redeemed and Moshiach would come, may it happen speedily in our days.
The last part of this aliyah contains the commandment of putting tzitzis on our four-cornered garments. This part of the aliyah is what we say for the third paragraph of the Shma Yisroel. On the subject of tzitzis, the Torah tells us that if we look at them, we will remember all of the commandments. On this, Rashi says that the gematria of the word tzitzis is 600, add that to the  threads, and the 5 knots and they total 613, the number of commandments.