In his book, In the Presence of Our Lord: The History, Theology and Psychology of Eucharistic Devotion,
Fr. Benedict Groeschel outlines "four kinds of prayer most appropriate in the presence of the Eucharist, namely adoration and praise, thanksgiving, repentance, and trusting intercession."

Here are suggestions for what to do during private Eucharistic Adoration:

• Pray the Psalms or the Liturgy of the Hours - Whether you are praising, giving thanks, asking for forgiveness, or seeking an answer, you’ll find an appropriate psalm. The ancient prayer of the Church called the Liturgy of the Hours presents an excellent way to pray through the Book of Psalms throughout the year.
• Recite the Jesus Prayer - Pray "Lord Jesus, have mercy on me, a sinner," repeatedly as you quiet your heart and mind.
• Meditate using Scripture - Choose a passage from the Bible. Read the words and ask God to let the passage speak to you. Pay special attention to anything that strikes you and ask God what He wishes for you to draw from that passage.
• Read the life of a saint and pray with him or her - Most holy men and women have had a great devotion to Our Lord in the Eucharist. Read about the saints and pray their prayers before the Blessed Sacrament.
• Pour out your heart to Christ and adore Him - Speak to Jesus, aware that you are in His presence, and tell Him all that comes to your mind. Listen for His response. Pray the prayer that St. Francis instructed his brothers to pray whenever they were before the Blessed Sacrament: "I adore you, O Christ, present here and in all the churches of the world, for by your holy cross you have redeemed the world."
• Ask for forgiveness and intercede for others - Think of those who have hurt you and request a special blessing for them. Ask God to forgive you for all the times you have neglected or hurt someone else. Bring before the Blessed Sacrament all those who have asked you to pray for them. Ask the Lord to address their concerns.
• Pray the Rosary - Saint John Paul II reminds us, "...is not the enraptured gaze of Mary, as she contemplated the face of the newborn Christ and cradled him in her arms, that unparalleled model of love which should inspire us every time we receive Eucharistic communion?" Ask Mary to join you as you gaze on Christ in the Eucharist and as you pray the Rosary.
• Sit quietly and just "be" in the presence of God - Think of a visit to the Blessed Sacrament as coming to see your best friend. Sit quietly and enjoy being in each other’s company. Instead of talking to the Lord, try listening to what He wants to tell you.