Only one passage speaks directly about the Spirit praying on behalf of the believer, but what a passage!
In the same way the Spirit also joins to help in our weakness, because we do not know what to pray for as we should, but the Spirit Himself intercedes for us with unspoken groanings. And He who searches the hearts knows the Spirit’s mind-set, because He intercedes for the saints according to the will of God (Romans 8:26-27).
Believers are not alone in their prayer burdens. When we do not know what to ask of God, the Spirit is there making intercession for us. Certainly, “the intense prayer of the righteous is very powerful” (James 5:16). But what a joy to know that the Holy Spirit intercedes for us! He knows the will of God far better than we do. True prayer goes from our hearts through the mind of the Spirit to the Son, who intercedes for us with the Father.
Therefore, since we have a great high priest who has gone through the heavens, Jesus the Son of God, let us hold firmly to the faith we profess. For we do not have a high priest who is unable to sympathize with our weaknesses, but we have one who has been tempted in every way, just as we are—yet was without sin. Let us then approach the throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need (Hebrews 4:14-16 NIV).
Two passages speak about the believer praying in the Spirit.
With every prayer and request, pray at all times in the Spirit, and stay alert in this, with all perseverance and intercession for all the saints (Ephesians 6:18).
But you, dear friends, building yourselves up in your most holy faith and praying in the Holy Spirit (Jude 20).
What does in mean to pray in the Spirit? Albert Barnes, says, “By the aid of the Holy Spirit; or perhaps it may mean that it is not to be prayer of form merely, but when the spirit and the heart accompany it. The former idea seems, however, to be the correct one.”
Paul wrote, “I will pray with the spirit, and I will also pray with my understanding” (1 Corinthians 14:15). If the apostle meant the Holy Spirit instead of his spirit, praying in the Spirit is best understood as praying under the Spirit’s influence and with His assistance. It does not mean praying in unintelligible tongues!