There is a vocal prayer of preparation in the Christian East that forms a type of introduction to other prayers within particular services. For example, priests and deacons pray this as a prologue to Divine Liturgy or as the preface to house blessings, memorials for the deceased, patronal feasts, the daily cycle of prayers and more. It can also be used to start your own prayers.
Notice that the beauty of this prayer is in its simplicity. It contains, among other things, the Prayer to the Holy Spirit, the Lord's Prayer, and the Trisagion Prayer (this extended prayer of preparation is also generally referred to as the Trisagion Prayers even though it contains much more).
Since these are familiar prayers, they provide a sense of comfort. They're also catechetical, which means they remind us of the basic tenets of faith. In the context of personal prayer, it would be natural to add your intercessory requests afterward. While this prayer isn't normally envisioned as a launching point for meditation, it is helpful to keep us focused on God as the center of our life.
"Glory to Thee, our God, Glory to Thee.
"O Heavenly King, Comforter, the Spirit of Truth, present in all places and filling all things, the Treasury of good things and Giver of life: Come, and abide in us, cleanse us from every sin, and save our souls, O Good One.
"Holy God, Holy Mighty, Holy Immortal: have mercy on us. (3 times).
"Glory to the Father, and to the Son, and to the Holy Spirit, both now and ever, and unto the ages of ages. Amen.
"All-Holy Trinity, have mercy on us. Lord, cleanse us from our sins. Master, pardon our iniquities. Holy God, visit and heal our infirmities for Thy name's sake.
"Lord, have mercy. Lord have mercy. Lord have mercy.
"Glory to the Father, and to the Son, and to the Holy Spirit, both now and ever, and unto the ages of ages.
"Our Father, who art in Heaven, hallowed be Thy name. Thy Kingdom come, Thy will be done, on earth as it is in heaven. Give us this day our daily bread, and forgive us our trespasses as we forgive those who trespass against us; and lead us not into temptation, but deliver us from the evil one.
"For Thine is the Kingdom, and the power, and the glory, of the Father, and of the Son, and of the Holy Spirit, now and ever and unto ages of ages. Amen."