Regarding levels of heaven and whether the Bible teaches it or not, the very first verse of Genesis 1 does reference more than one when it states the following.
In the beginning God created the heavens and the earth (Genesis 1:1, HBFV).
Although the King James translation uses the word "heaven" in the first verse of Genesis, most modern Bibles correctly translate this verse as it is listed above. But does God's word teach a single spiritual realm that has levels or were multiple places created?
We can begin to track down the answer by looking further down in Genesis 1. Verses 7 and 8 are the first place a specific spiritual domain is mentioned.
And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament; and it was so. And God called the firmament heavens (Genesis 1:7 - 8).
This celestial location is the atmosphere that surrounds the earth, the place where clouds exist and birds, along with jets, fly in every day. Jeremiah confirms where this place is located in the fourth chapter and verse 25. The book of Revelation reaffirms this fact when it says, "Then I saw an angel standing in the sun; and he cried out with a loud voice, saying to all the birds that fly in the midst of heaven . . ." (Revelation 19:17).
The second place the Bible says God created can be found in Genesis 22, where it states that Abraham's descendants would be as "the stars of the heaven" (Genesis 22:17). This is where our moon, the sun, the planets in our solar system, distant stars and galaxies, and the rest of the universe is located.
Where, you might wonder, does God's throne exist? The Lord's throne cannot be in the first place mentioned above (the sky) since God does not yet dwell with mankind. Could it be in the second, what we call space? Job 22 sheds a little more light on the subject when he writes "Is not God in the height of heaven? And see the highest stars, how lofty they are!" (Job 22:12).
This suggests an additional domain that is even higher than the stars. It is the Apostle Paul who reveals and confirms the existence of a spiritual realm. Paul, speaking indirectly about what occurred to him, states that somehow, someway, he was transported to a third spiritual realm (2Corinthians 12:2 - 4) where he heard things he could not repeat to others!
Note that Paul, in 2Corinthians 12, is not stating there are levels of heaven. He is, however, stating he was taken to the third of a sequence of three (logically implying that a first and second domains exist).
Distinctions
God tells us in His word that He indeed created three separate and distinct (not necessarily levels) heavenly places.
1) The atmosphere (sky) that surrounds the earth
2) The Universe or what we call Space
3) The place where His throne is located
The concept that the Bible teaches there are levels of heaven highly likely originated with the Middle Ages' poem entitled "Divine Comedy" by Dante (written somewhere between 1308 - 1321). This writing describes, in graphic detail, his supposed "travels" through the realms of Purgatory, Hell and others, each of which is comprised of nine parts or sections.