Great question! So many Christians don't think about how the Trinity shapes their prayer life, sadly. I'm glad this is a concern for you!
It seems that Jesus provides a general pattern for speaking to God in the Lord's Prayer where we see that the Father is the one to whom we primarily address our payers (Matt. 6:9). Elsewhere Jesus says that we should pray in his own name (John 14:13). One of the main ministries of the Holy Spirit is to empower the church -- what better way to do it than with prayer (Eph. 6:18)? Taking this together, it seems that a basic way to pray is to the Father, in the power of the Spirit, and in the name of the Son.
Does this pattern preclude praying to any of the three persons individually? I don't think so. The reason for this is that the three persons are equally God -- theologians explain that each person "subsists" in the one divine essence of the Godhead. Because of this, they should equally be worshiped and prayed to. For instance, the three share in the divine name that Jesus says disciples should be baptised into in the Great Commission (Matt 28:19). We see indications of praying to Jesus in texts like Thomas' great exclamation of "My Lord and my God!" when he encountered the post-resurrection Christ, or the prayer of Paul in 1 Cor. 16:22 when he says "Maranatha!" or "Come Lord Jesus!" It is entirely appropriate to address the Spirit because one of his main roles is to aid our prayers; indeed, he intercedes for us when we have trouble praying (Rom. 8:26-27).
With that said, I would also encourage you to be thoughtful in the way that you address each person. Sometimes Christians don't think through the roles of the particular persons of the Godhead and will misappropriate them. For instance, don't thank the Father for dying on the cross! The Son died! To say that the Father died is to commit the error of "Patripassianism." Of course, the Father is deeply involved in our redemption, as is the Son and Spirit, but each in a special way. So it might be best to pray: "Father, thank you for sending your Son. Jesus, thank you for being willing to die in my place. Holy Spirit, thank you for raising Christ from the dead!"