There is so much to be written about the plan of salvation, but my hope in writing this is to concisely summarize John 3 and the new birth experience.
John 3:1-3
1 There was a man of the Pharisees, named Nicodemus, a ruler of the Jews:
2 The same came to Jesus by night, and said unto him, Rabbi, we know that thou art a teacher come from God: for no man can do these miracles that thou doest, except God be with him.
3 Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God.
In the first few verses of John 3, we read about a man named Nicodemus approaching Jesus and recognizing Jesus' authority. Jesus doesn't beat around the bush, and He immediately jumps into the topic of being "born again." In fact, Jesus goes so far as to say that unless you be born again, you cannot see the kingdom of God, which would be Heaven.
Seeing (or entering) the kingdom of God would be salvation, becoming saved. And there is a neatly drawn line in the sand that one must cross before entering into the kingdom of God. Jesus conveniently spells it out for us: in order to enter into the kingdom of God, we must be born again.
So we must be born again to enter the kingdom. But how exactly are we born again? Nicodemus asked this very question in the next verse.
John 3:4
4 Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother's womb, and be born?
Nicodemus is understandably confused, as he is viewing things in a natural, physical way. But Jesus talks about being born again in a spiritual manner.
John 3:5-6
5 Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.
6 That which is born of the flesh is flesh; and that which is born of the Spirit is spirit.
When we are born as an infant, it is a fleshly birth. When we are born again (of water and Spirit), it is a spiritual birth.
Let's look at one thing at a time, first water, then Spirit.
What does it mean to be born of the water? Simply put, this means to be baptized in Jesus name. There are so many examples in the book of Acts of people being baptized, for conciseness sake I will only list a few.
Acts 2:38
38 Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.
Acts 10:48
48 And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days.
In both the preceding verses, Peter commands people to be baptized, specifically baptized in Jesus name. The Greek word used for baptism in the Bible is baptizō, which literally means "to overwhelm with water, to immerse, to submerge." With all this in mind, hopefully we can see that to be born of the water = water baptism in Jesus name.
How are we born of the Spirit?
First off, it is worth mentioning that the Holy Ghost and Holy Spirit are the same and used interchangeably. In fact, the word Ghost and Spirit in the Bible both come from the same Greek word pneuma, which means "a current of air, breeze, spirit, ghost." It is where our modern word pneumatic comes from, such as air-powered pneumatic tools.
Jesus actually gives us some evidence of being born of the Spirit in our main text of John 3.
John 3:7-8
7 Marvel not that I said unto thee, Ye must be born again.
8 The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit.
It is important to note that the word "wind" in the above verse is the same Greek word pneuma for Spirit. So we could quite literally read the first part of the verse like this (with modern interpretation for readability):
"The SPIRIT blows where it wishes to go, and you hear the sound of it."
Just like in the natural world, we cannot see the wind, but we can certainly hear the wind. The same applies to the Holy Spirit: We cannot see the Spirit, but we can certainly hear Him!
Jesus said "so is everyone that is born of the Spirit." He is specifically referring to the SOUND. When anyone is born of the Spirit, you can hear the SOUND.
The sound would be speaking in other tongues.
To illustrate this, let's examine when Peter was preaching to Gentiles, who up to this point hadn't been preached to since the Jews were only preaching amongst themselves.
Acts 10:44-46a
44 While Peter yet spake these words, the Holy Ghost fell on all them [Gentiles] which heard the word.
45 And they of the circumcision which believed [Jews] were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost.
46a For they heard them speak with tongues, and magnify God.
The Jews were astonished because they had thought the new birth experience was only for them, but when Peter preached to the Gentiles, the Holy Ghost fell on them and they received the Holy Ghost.
How did the Jews know that the Gentiles received the Holy Ghost? They heard them speak with tongues. They heard the SOUND. Hopefully we can see that being born of the Spirit = receiving the Holy Ghost. And we can tell if someone receives the Holy Ghost by the sound they make, specifically the sound of speaking in tongues.
Acts 10:46a-47
46b Then answered Peter,
47 Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?
Are you born of the water and of the Spirit?