Yes, the Bible does teach that some day God will give us new heavenly bodies -- bodies that will never wear out or get sick. The Bible says that "we will all be changed -- in a flash, in the twinkling of an eye, at the last trumpet" (1 Corinthians 15:51-52).
Can we understand exactly how God is going to do this? No, not fully (which is why the Bible calls it a mystery). But one thing is clear: We don't need to worry about this -- because nothing is beyond God's ability. If it's necessary at the end of time for Him to bring together all the scattered molecules that made up our original bodies, He can do it -- because He is God, and He is all-powerful. However, He may use some other method. God created Adam from the dust of the earth, and He can re-create us from the earth's dust also (see Genesis 2:7).
Questions like this may be interesting -- but don't let them keep you from seeing what's really important. What's really important is this: Because of Jesus Christ we have hope -- hope for our lives right now, but hope most of all for life beyond the grave. This world is not all! Ahead of us is heaven if we know Christ -- and some day we will be with Him forever.
Are you ready for that day? Make sure, by turning to Christ and trusting Him alone for your salvation. He died and rose again to give you a new life -- beginning today.