I have just started reading Genesis and have a question about Noah's family. If Noah and his family were the only people left on earth after the flood, why did most of the world forget God and worship other gods, like in Asia and Africa? After being blessed and allowed to live by God, I would think that Noah's family would make sure God's story was passed down through all generations (and parts of the world).
The same way it was lost in the first place.
For the same reason that Cain killed Abel.
Noah was saved physically from the physical calamity, but he was not regenerated. He was not yet saved from sin. He was still a sinful creature as were the other members of his family. The promise from Genesis 3:15 was not yet fulfilled.
The Bible and indeed all of human history is full of the same story. It does not really matter how earth shattering a miracle is witnessed or how clearly people know that God did something (up to and including saving their hides), we all still turn away from God. It's a heart problem.
In Exodus you will find a people that in the course of one lifetime saw God repeatedly intervene on their behalf and they repeatedly turned against him.
The message of the Bible is that you and I are that people.
The solution is still forthcoming. It is found in Christ, but it's a work in progress—secured and completed at the cross but still being revealed to us today. The destruction of the corrupt earth in Noah's day and Noah's being brought over into a new creation was a TYPE or pattern of salvation: a symbol of how the real thing would be but it was not the real thing. You will find in the New Testament that the flood story is a kind of baptism.
1 Peter 3:20-21 (ESV)
20 because they formerly did not obey, when God's patience waited in the days of Noah, while the ark was being prepared, in which a few, that is, eight persons, were brought safely through water. 21 Baptism, which corresponds to this, now saves you, not as a removal of dirt from the body but as an appeal to God for a good conscience, through the resurrection of Jesus Christ,
Just as baptism is a physical symbol of being cleansed and the reality that we will be resurrected with Christ yet it does not make us perfect or stop us from sinning, so the flood was a symbol of the same thing but it did not fix the heart problem. It didn't end the sin-nature of man.
One day, the destruction and re-creation that was foreshadowed in the flood (and in baptism) will be completed for real.
Matthew 24:37 (ESV)
37 For as were the days of Noah, so will be the coming of the Son of Man.
The corrupt wicked world that has rejected God will be judged and destroyed and God's people—the ones he chooses to save—will be brought into a the new heavens and new earth only this time with a our natures renewed to what Adam's should have been. Only then will there be a world that is imperishable and incorruptible.
What you expected to have happen after the flood will happen, but the flood was only a precursor to the real thing.
Very, very good question.
While the majority of man tended toward wickedness after the flood, faith was not lost, but was a possession of a very small minority.
After the flood there was once again a quick division between the righteous (like Noah) and the unrighteous (like his son Ham and Nimrod). From the exit of the ark, righteous ones from Noah ones up to Abraham and beyond were the ones entrusted with God's promises and indeed presence throughout their lives, and progressively that heritage was allowed for all of the world to benefit from.
Your question actually reflects one of the central themes of the bible which is the progression of the seeds of wickedness VS righteousness.
Believe it or not, both good and bad people exited the ark!
It manifested in a simple matter of morality, which ended with Noah saying this:
25 He exclaimed, Cursed be Canaan! He shall be the [a]servant of servants to his brethren!
26 He also said, Blessed be the Lord, the God of Shem! And blessed by the Lord my God be Shem! And let Canaan be his servant.
Now read Genesis 11:1-10 regarding the first large-scale rebellion against God. Josephus even states that the "bitumen" they used to seal the tower was a countermeasure against another flood (which God promised he would not do again...the very construct of this tower was based on a lack of faith and trust in God...).
The key part of your answer though (and its a broad subject) is answered in the circumstances of the Tower of Babel which was the first organized large-scale endeavor against God's will, headed by a man who himself was known as a contender against God.
God had told them to spread out, whereas the people (headed by Nimrod) decided to build a tower and make a "name for themselves". You can expect much, much knowledge was concentrated here - practical, historical, and so forth. Yet notice that the general leanings of man were against God's will, especially in this instance, harmonizing with what God said shortly after the flood:
...I will never again curse the ground because of man, for the imagination (the strong desire) of man’s heart is evil and wicked from his youth; neither will I ever again smite and destroy every living thing, as I have done.
So upon going down to "confuse the languages" of man forcing him to abandon his rebellious project, many, many centralized teachings and ideas (some even involving a global flood, among many other religious themes) were carried all over the earth. That is why you can see harmony between things like the Egyptian pyramids and the Aztec structures, as well as harmonies between many pagan religions. They all stemmed from this central act of opposition.
Later on this "Babel" would extend it's rebellious lineage into pagan and violent civilizations such as Nineveh and Babylon, which extends even further into the prophetic "Babylon the Great" which acts as a chief contender against God, the same as at it's birth after the flood.
With all of the rebellion of the post-flood events, there was always a righteous descendent of Noah, the first most notable of which was Abraham, the "father of all having faith".
Faith was lost because; the people disobeyed God, did not heed the prophecies of Enoch, and ignored Noah's preaching.
Woe unto them! for they have gone in the way of Cain, and ran greedily after the error of Balaam for reward, and perished in the gainsaying of Core. 14 And Enoch also, the seventh from Adam, prophesied of these, saying, Behold the Lord cometh with ten thousands of his saints,
These are the generations of Noah: Noah was a just man and perfect in his generations, and Noah walked with God.
II Peter 2;5
And spared not the old world, but saved Noah the eighth person, a preacher of righteousness, bringing in the flood upon the world of the ungodily;
Ezra 10;11 Now make confession unto the Lord God of your fathers, and do his pleasure: and seperate yourselves from the people of the land, and from strange wives.
And it came to pass, when men began to multiply on the face of the earth, and daughters were born unto them, 2 That the sons of God saw that the daughters of men that they were fair; and took them wives of all which they chose.
And God saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually. 11 The earth was also was corrupt before God, and the earth was filled with violence.
And the Lord said unto Noah, Come thou and all thy house into the ark; for thee have I seen righteous before me in this generation.
And I will make thee swear by the Lord, the God of heaven, and the God of the earth, that thou shalt not take a wife unto my son of the daughters of the Canaanites, among whom I dwell: 4 But thou shalt go unto my country, and to my kindred,and take a wife unto my son Isaac.
Behold, I stand here by the well of water; and the daughters of men of the city come out to draw water:15 And it came to pass, before he had done speaking, that behold, Rebekah came out,..........
Proverbs 24;1 Be not envious of evil men, neither desire to be with them.
They shall not dwell in thy land, lest they make thee sin against me: for if thou serve their gods, it will surely be a snare unto thee.
II Corinthians 6;14 Be ye not unequally yoked together with unbelievers: for what fellowship hath righteousness with unrighteousness? and what commuion hath light with darkness?
Already some good answers here, but I take issue with the premise of your question:
If Noah and his family were the only people left on earth after the flood
This is one possible interpretation of the flood story, but it's not the only one. There are people who think that the flood was local.
This would answer the 'how come there are different religions all over the world?' part of your question.
The 'more local' part of your question: how could the descendants of the people who LIVED this 'salvation' abandon the God who saved them? is what the other answers here are addressing. You say you are just starting to read Genesis. In case this is the first time, and you are planning on continuing through the Old Testament, you will see that this is the recurring theme: mankind screwing up despite (sometimes) their best efforts, and God alternately chastising and finding a way to make things right again.