So what should a person do when they want to give their lives to Christ?
If someone approaches you wanting to do this, in my observance so far of scripture, simply tell them of their need to repent, as John did, and of their need to believe in Christ as the Son of God, sent from heaven to die for our sins, so we can be connected with God again, and their need to confess their sins before Christ.
When viewing the scriptures together, they all point to this.
Putting someone on the spot to give their hearts to God now, with you, and leading them in a repeat after me prayer, and then telling them that because they said that prayer they are now saved and born again, from my observance of scripture, should not be done.
We are not born again because of a special prayer that is repeated. We are born again and enter in to God's kingdom when we repent and confess our sins, and believe in Christ Jesus who gave His life so we can have everlasting life.
Do we believe what Jesus said?
Joh 6:40 "This is the will of my Father who sent me, that everyone who sees the Son, and believes in him, should have eternal life; and I will raise him up at the last day.”
So, how does prayer fit in? Well, any conversation between us and God is prayer. We talk to Him and He talks to us. Prayer is not a one sided communication with God.
This is all I have read being preached: repent, confess, believe in Christ and you will be saved, or have everlasting life.
I think the same applies when sharing the word to a group or crowd, and the Holy Spirit prompts to call those present to repentance, then simply do so as John and Peter did, and share the need to repent, confess their sins, and believe (committing yourself fully) in Christ.
See, we need to trust God that that person was sincere in their repentance, confession, and believing in Christ. We can't see their hearts.
However, the fruit of repentance will be seen immediately if that person truely repented. (Matthew 3:8)
Then, after being born again, that person follows Christ through His word and the Holy Spirit leading. You can't just believe in Christ truely and not follow Him. Because the Apostles believe Christ,  they followed Him. (John 6: 68 & 69)
Another important thing that must follow accepting Christ is discipleship. We must make sure the new believer understand what happened to him/her, and be taught the requirements Christ has to follow Him.
May we walk as Christ teaches us to.