OK, here's some verses on the matter.
2 Corinthians 5:17 (New International Version)
17 Therefore, if anyone is in Christ, he is a new creation; the old has gone, the new has come!
Galatians 6:15 (New International Version)
15 Neither circumcision nor uncircumcision means anything; what counts is a new creation.
John 3:5 (New International Version)
5 Jesus answered, "I tell you the truth, no one can enter the kingdom of God unless he is born of water and the Spirit.
Luke 17:21 (New International Version)
21 nor will people say, 'Here it is,' or 'There it is,' because the kingdom of God is within you."
John 3:7 (New International Version)
7 You should not be surprised at my saying, 'You must be born again.'
Romans 8:29 (New International Version)
29 For those God foreknew he also predestined to be conformed to the likeness of his Son, that he might be the firstborn among many brothers.
Colossians 1:15 (New International Version)
15 He is the image of the invisible God, the firstborn over all creation.
Colossians 1:18 (New International Version)
18 And he is the head of the body, the church; he is the beginning and the firstborn from among the dead, so that in everything he might have the supremacy.
Hebrews 12:23 (New International Version)
23 to the church of the firstborn, whose names are written in heaven. You have come to God, the judge of all men, to the spirits of righteous men made perfect
John 6:53-54 (New International Version)
53 Jesus said to them, "I tell you the truth, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. 54 Whoever eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day.
So what is it to be "born again" a "new creation"?
2 Corinthians 5:17-21 (New International Version)
17 Therefore, if anyone is in Christ, he is a new creation; the old has gone, the new has come! 18 All this is from God, who reconciled us to himself through Christ and gave us the ministry of reconciliation: 19 that God was reconciling the world to himself in Christ, not counting men's sins against them. And he has committed to us the message of reconciliation. 20 We are therefore Christ's ambassadors, as though God were making his appeal through us. We implore you on Christ's behalf: Be reconciled to God. 21 God made him who had no sin to be sin for us, so that in him we might become the righteousness of God.
If you are reborn, you are a new creation, "born from above" (see the NIV footnote John 3:3) spiritually alive to God, reconciled to the Father through Jesus, with your sins not counted against you.
Yes, you still have "the flesh" which will die one day, along with its sinful desires.
John 11:25-26 (New International Version)
25 Jesus said to her, "I am the resurrection and the life. He who believes in me will live, even though he dies; 26 and whoever lives and believes in me will never die. Do you believe this?"
Romans 8:13 (English Standard Version)
13 For if you live according to the flesh you will die, but if by the Spirit you put to death the deeds of the body, you will live.
So what happens when you are "born again"?
Ephesians 1:13-14 (New International Version)
13 And you also were included in Christ when you heard the word of truth, the gospel of your salvation. Having believed, you were marked in him with a seal, the promised Holy Spirit, 14 who is a deposit guaranteeing our inheritance until the redemption of those who are God's possession—to the praise of his glory.
You believe the gospel and the Holy Spirit comes to live within you.
Does one need to have "an experience"?
No, it comes by faith Hebrews 11
Can one have "spiritual experience"?
Certainly Acts 2
One important point to note is that "baptism of the Spirit" is another experience which possible might happen at the time of being born again but quite often some time after or not at all. That said, Spirit baptism (not to be confused with water baptism) is for all who believe!
Acts 19:1-7 (New International Version)
Paul in Ephesus
1 While Apollos was at Corinth, Paul took the road through the interior and arrived at Ephesus. There he found some disciples 2 and asked them, "Did you receive the Holy Spirit when you believed?"
They answered, "No, we have not even heard that there is a Holy Spirit."
3 So Paul asked, "Then what baptism did you receive?"
"John's baptism," they replied.
4 Paul said, "John's baptism was a baptism of repentance. He told the people to believe in the one coming after him, that is, in Jesus." 5 On hearing this, they were baptized into the name of the Lord Jesus. 6 When Paul placed his hands on them, the Holy Spirit came on them, and they spoke in tongues and prophesied. 7 There were about twelve men in all.
Are there other "spiritual experiences" outside of the Kingdom which are like being born again or baptism in the Spirit?
Certainly, there are plenty of counterfeits out there, many which will give you a "feel good" factor for a time. However, look for the fruit. Any lasting peace? Any lasting joy? Any nasty side effects?
Blessings,
Stephen