Before you and I were here on Earth as we are today, we were together. We hung out together, you and I, as spirit children of our Heavenly Father. If you don't believe me, I promise you it's in the scriptures.
And it makes sense. This life of ours has a greater purpose than you suppose.
Anyway, I find it interesting that good and evil have been at battle since then.
Our loving Heavenly Father, God, proposed this glorious plan for us to come to Earth, to gain bodies, to learn, to have families, and to find our way by choosing for ourselves.
Our elder brother volunteered to be our Saviour, because one would be needed. And Lucifer threw a tantrum.
And the war in heaven began.
And the battle between choosing the right and being swayed to do evil continued. It followed us down. We are human, after all.
Adam and Eve are great role models for how parents should be, and how families can be.
That is, we want the best for our children, and we teach them all we can so they can choose good and reach their potential.
Cain didn't do so well with that. But nonetheless, it's good to know families have been that way since forever. No family is perfect. Whatever perfect is. No one has it all together.
The scriptures can assure us of that.
Read here for an example of righteousness in Adam's posterity.
Read here for an example of wickedness in Adam's posterity.
Before moving on to the next chunk in biblical history, I think it important to note the covenant that God made after the Flood. Remember, the wicked had just been destroyed by flood, and only a remnant of mankind was preserved.
And God spake unto Noah, and to his sons with him, saying,
And I, behold, I establish my covenant with you, and with your seed after you;
And with every living creature that is with you, of the fowl, of the cattle, and of every beast of the earth with you; from all that go out of the ark, to every beast of the earth.
And I will establish my covenant with you; neither shall all flesh be cut off any more by the waters of a flood; neither shall there any more be a flood to destroy the earth.
And God said, This is the token of the covenant which I make between me and you and every living creature that is with you, for perpetual generations:
I do set my bow in the cloud, and it shall be for a token of a covenant between me and the earth.
And it shall come to pass, when I bring a cloud over the earth, that the bow shall be seen in the cloud:
And I will remember my covenant, which is between me and you and every living creature of all flesh; and the waters shall no more become a flood to destroy all flesh.
And the bow shall be in the cloud; and I will look upon it, that I may remember the everlasting covenant between God and every living creature of all flesh that is upon the earth.
And God said unto Noah, This is the token of the covenant, which I have established between me and all flesh that is upon the earth.
(source - some good thoughts about the rainbow here)
The rainbow was the token, or symbol, of the covenant made - that God would never again flood the Earth to destroy all living things.
I do recommend noting Joseph Smith's translation of this passage in Genesis here (JST, Genesis 9:17–25).
But the following may also shed more light on this token and the covenant it signifies:
“The Lord hath set the bow in the cloud for a sign that while it shall be seen, seed time and harvest, summer and winter shall not fail; but when it shall disappear, woe to that generation, for behold the end cometh quickly.”I find it curious that this event is not believed by some.
“I have asked of the Lord concerning His coming; and while asking the Lord, He gave a sign and said, ‘In the days of Noah I set a bow in the heavens as a sign and token that in any year that the bow should be seen the Lord would not come; but there should be seed time and harvest during that year: but whenever you see the bow withdrawn, it shall be a token that there shall be famine, pestilence, and great distress among the nations, and that the coming of the Messiah is not far distant.’” 
But hey, this was not believed by the great and wise people of Noah's day either ..
And just like that, the Earth was given another chance. A clean state.
Hmm. And then what happened ... ?
Some clever people decided to build a tower to get to heaven.
And the whole earth was of one language, and of one speech.
And it came to pass, as they journeyed from the east, that they found a plain in the land of Shinar; and they dwelt there.
And they said one to another, Go to, let us make brick, and burn them throughly. And they had brick for stone, and slime had they for morter.
And they said, Go to, let us build us a city and a tower, whose top may reach unto heaven; and let us make us a name, lest we be scattered abroad upon the face of the whole earth.
Why on earth did they do that??
What were their motives? How could they have gotten it so wrong?
Well hang on a tick .. let's look at what happened as a result of this silliness.
Do you remember what the consequences of their actions were? Have a look in Genesis again:
And the Lord came down to see the city and the tower, which the children of men builded.
And the Lord said, Behold, the people is one, and they have all one language; and this they begin to do: and now nothing will be restrained from them, which they have imagined to do.
Go to, let us go down, and there confound their language, that they may not understand one another’s speech.
So the Lord scattered them abroad from thence upon the face of all the earth: and they left off to build the city.
Therefore is the name of it called Babel; because the Lord did there confound the language of all the earth: and from thence did the Lord scatter them abroad upon the face of all the earth.
And so, people were scattered throughout the Earth, all talking gibberish.
This account, other than providing an explanation for all the languages people speak nowadays (Pig Latin and such probably not included), gosh we learn a lot about how quickly man forgot the lessons of the Flood and turned again from the Lord.
It is the Book of Mormon - the book we should be reading side by side with the Bible - that offers more detail about the confounding of languages, which may not have been so instantaneous. I think so, because we read about Jared and his brother, who prayed to the Lord and asked that their language not be confounded. And then he asked that his friends be spared the gibberish. You can read all about it here, in the book of Ether.
 Smith, Teachings, p. 305, 340–41