No, There are not different levels of heaven. When you're in, you're in. There may be different levels of reward, but the Bible never tells us that there are different levels of heaven.
However, the Bible does speak of the "heavens" (plural) in different ways. The "third heaven" is said to exist in 2 Cor. 12:2, which says, "I know a man in Christ who fourteen years ago—whether in the body I do not know, or out of the body I do not know, God knows—such a man was caught up to the third heaven." It is logical to conclude that if there is a third heaven, there is also a first and second. But, Scripture does not describe specifically the first and second heaven by mentioning them as such. Instead, we can see how the Bible speaks of different heavens by looking at scripture.
The first heaven apparently refers to the atmospheric area of the birds (Hosea 2:18) and clouds (Dan. 7:13). The second heaven is the area of the stars and planets (Gen. 1:14-18) and some say is also the abode of all supernatural angelic beings. The third heaven is the dwelling place of the triune God. Its location is unrevealed. (See Matt. 23:34-37; Luke 10:20; and Rev. 22:2, 20-27).