What is the gospel, and how and when does a person come to obey the gospel?

The gospel is "…How that Christ died for our sins according to the scriptures; and that he was buried, and that he rose again the third day according to the scriptures:" - 1 Corinthians 15:3-4 KJV

A person obeys the gospel of Jesus’ death, burial, and resurrection, by repentance – which is death to sin, by water baptism in Jesus name – which is burial of old nature, and by Holy Ghost infilling – which is resurrection to new life. (Romans 6:1-6)

On the day of Pentecost when the church first began, Peter preached the gospel – the death, burial, and resurrection of Jesus Christ. When he got to the end of his message, the Bible says, "Now when they heard this, they were pricked in their heart, and said unto Peter and to the rest of the apostles, Men and brethren, what shall we do?" - Acts 2:37 KJV

"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. For the promise is unto you, and to your children, and to all that are afar off, even as many as the Lord our God shall call." - Acts 2:38-39 KJV

"And with many other words did he testify and exhort, saying, Save yourselves from this untoward generation." - Acts 2:40 KJV

When did they get baptized in Jesus name after they heard the preaching of the gospel?

It was the same day.

"Then they that gladly received his word were baptized: and the same day there were added unto them about three thousand souls." - Acts 2:41 KJV

When God opened the prison doors for Paul and Silas, the keeper of the prison was about to kill himself, supposing that the prisoners had escaped. But Paul cried, Do not harm yourself - we are all here! Trembling, the prison keeper said, Sirs, what must I do to be saved? (Acts 16:23-30)

"And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house. And they spake unto him the word of the Lord, and to all that were in his house." - Acts 16:31-32 KJV

When Paul and Silas preached unto the jailor and all that were in his house the word of the Lord, they all wanted to be baptized in Jesus name.

When did the prison keeper and his household get baptized? The same hour of the night.

"And he took them the same hour of the night, and washed their stripes [their wounds]; and was baptized, he and all his, straightway." - Acts 16:33 KJV

After Phillip preached Jesus unto the Ethiopian eunuch, the man asked Phillip if he could be baptized.

When did Phillip baptize the man?

"And as they went on their way, they came unto a certain water: and the eunuch said, See, here is water; what doth hinder me to be baptized?" - Acts 8:36 KJV

"And Philip said, If thou believest with all thine heart, thou mayest. And he answered and said, I believe that Jesus Christ is the Son of God. And he commanded the chariot to stand still: and they went down both into the water, both Philip and the eunuch; and he baptized him." - Acts 8:37-38 KJV

After Saul was blinded by a great light, he heard the voice of Jesus speaking to him.

"And I said, What shall I do, Lord? And the Lord said unto me, Arise, and go into Damascus; and there it shall be told thee of all things which are appointed for thee to do." - Acts 22:10 KJV

God sent Ananias to Saul.

"And Ananias went his way, and entered into the house; and putting his hands on him said, Brother Saul, the Lord, even Jesus, that appeared unto thee in the way as thou camest, hath sent me, that thou mightest receive thy sight, and be filled with the Holy Ghost." - Acts 9:17 KJV

"And immediately there fell from his eyes as it had been scales: and he received sight forthwith, and arose, and was baptized." - Acts 9:18 KJV

Later in the book of Acts, Paul was testifying of his vision and the voice of Jesus, and how Ananias came to him, to heal his blindness and tell him how God had chosen him.

"For thou shalt be his witness unto all men of what thou hast seen and heard." - Acts 22:15 KJV

Paul then told how Ananias told him to be baptized.

"And now why tarriest thou? arise, and be baptized, and wash away thy sins, calling on the name of the Lord." - Acts 22:16 KJV

In Acts 10, God sent Peter to preach to Cornelius and his household, which were gentiles. While Peter preached the gospel to them, God filled them with His Holy Ghost.

When Peter saw that the Gentiles received the Holy Ghost (they began speaking in tongues), he then commanded them to be baptized.

"Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we? And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days." - Acts 10:47-48 KJV

When Paul came to Ephesus, he found disciples of John the Baptist. When Paul asked them how they were baptized, they told him “Unto John's baptism.”

"Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus." - Acts 19:4 KJV

When they heard they needed to be baptized in Jesus name, they were then baptized!

"When they heard this, they were baptized in the name of the Lord Jesus." - Acts 19:5 KJV

"And when Paul had laid his hands upon them, the Holy Ghost came on them; and they spake with tongues, and prophesied." - Acts 19:6 KJV