These verses in Romans 8:18-30 mention three things that are or will be groaning: creation, the believer and the Spirit. The creation groans because of the affect of sin. Creation was created by God and was originally called "good" by God. But, now has experienced the groans and pains of childbirth, waiting until the glory that God will reveal when he will one day return. Notice though that verse 18 tells us that whatever suffering we are undergoing now is not to be compared to the glory that will be revealed.
I've wondered to myself if those who talk about global warming have ever researched these verses. Pollution and toxins in the air are increasing and should be our concern because we are stewards of the world around us. God gave us that responsibility as humans in the garden of Eden. But, there is an effect on creation that exists even more than pollution or coal ash spills and it is sin. The good news is that God is going to return one day and undo all the damage that sin and the effects of it have done on His creation. In the meantime, christians should be at the forefront of doing what we can such as recycling to help the environment.
The second groaning is that of the believer. In 2 Corinthians 5:1-9 it describes the body as a tent. A tent has a lot of limitations and is temporary. Our bodies are also not eternal. As we get older, it tends to wear down and cause us pain. We bear an image, Paul says in 1 Cor 15:49 of the earthly, but we shall bear an image that is heavenly. Part of the groaning is that our spirit longs for that day. That day when sin and death will be no more and God will wipe away all tears Revelations 21:4. One gret verse that I found is in 2 Corinthians 4:17-18 where it tells us that the light affliction that we now suffer is producing in us something of great weight of glory. It isn't something seen, but rather it is unseen. For what is seen is temporary, but what is permanent is unseen.
The third groaning is the Spirit, which intercedes for us. This should encourage us because the Spirit always knows the will and mind of God. His prayers are always answered because it is always according to His will. God can bring about good even when bad things happen. We should remember that we are called, justified and glorified. I like how the word "glorified" is not in future tense here, but is a present tense reality.
Related verses Col 3:4; Titus 2:13; 1 John 3:2; Jer 12:4,11, John 14:16, 17:22; 2 Cor 4:17.