There are many Christians who do accept repentance as a part of salvation (as well they should), but they view repentance as only "a change of mind" (from the Greek word metanoia). Repentance is not only a change of mind, I think, but it's also a change of direction that results in a change of action -- not merely an acknowledgement of sin, but also a turning from sin (and self) and to God.
Repentance is a complete surrender of the heart, mind, and will (a supernatural work of God). It's a proper response to the conviction of sin. That said, repentance is NOT "cleaning up yourself first" before you can come to God for saving. Not only is that impossible, but that is works and works cannot save anyone! (See: Are good works enough? 8 Reasons Why We Can't Work Our Way To Heaven) I like what one preacher said: "The call to Repentance in salvation is not 'clean up your act first for Jesus to accept you and save you.' You don't get good to get saved; you get saved to get good. Repentance is a deep conviction that you can no longer continue on the road that you are on, a road of destruction...a road of self rule or self righteousness, which leads to eternal ruin. And with that conviction, you turn to Christ in faith believing that He is the only Savior able to set you free from the penalty and power of sin through His substitutionary death and resurrection from the grave."
In short, God brings the sinner to Himself through repentance and faith in Jesus Christ in an initial act known as justification (the sinner is made right with God). Subsequent to justification is sanctification (the process of becoming more like Christ); this is a continual process throughout the earthly life of the believer. All of this is by God's grace. It is God's grace that saves and it is God's grace that sanctifies!
[There is no repentance without faith and there is no faith without repentance. Faith without repentance (easy-believism) is a false gospel and only produces false converts (See: False Converts Need to Be Warned and How Does the Bible Define a False Convert?).]
Genuine saving faith is always accompanied by repentance. A true saving faith is penitent faith (repentant faith): Godly sorrow. A false non-saving "demon" faith is impenitent faith (unrepentant faith): worldly sorrow. Godly sorrow (contrition) over sin produces a broken and contrite heart which compels the sinner to apologize to God and plead for mercy. In contrast, worldly sorrow (attrition) over sin is really a "non-apology" to God and more of a "sorry I got caught" mentality and maybe a hatred for the consequences of sin, but not necessarily the sin itself. Worldly sorrow over sin leads to death. Only Godly sorrow over sin produces repentance that leads to salvation: Life. (See: Godly Sorrow, Saving Faith vs Demon Faith, and also Mike Gendron's presentation below).
"The Greek-English Lexicon provides an excellent definition of the common Greek word 'metanoia' used for repentance in the New Testament. It means 'to change one's way of life as the result of a complete change of thought and attitude with regard to sin and righteousness.' We find in its meaning a conviction that produces action. Such a change is profoundly noted by Paul's description of the Thessalonians: 'You turned to God from idols to serve a living and true God' (1 Thes. 1:9). These saints displayed three elements of repentance: a turning from sinful behavior, a turning to God and a desire to serve Him. It is impossible to turn to God without turning away from sin. People who continue to live in habitual sin and think they are saved must be warned that Jesus saves us from our sin, not in our sin."
"In the Gospel of Jesus, men are called to repent of their sins and believe the Gospel. Those who obey the call are promised eternal life. The rest are warned that they will perish under the wrath of God if they continue in their unrepentant and unbelieving state. In the Gospel of Paul, the very same promises and warnings are given. The apostle solemnly testified to both Jews and Greeks of repentance toward God and faith in our Lord Jesus Christ. He proclaimed that God has commanded all people everywhere to repent, and he warned men not to be deceived by empty words, for the wrath of God is coming upon the disobedient."
"Never forget that acknowledging the existence of God doesn’t mean an entrance into Heaven, but (as the Bible so clearly teaches), repentance and faith in Jesus Christ, does."
[The Bible says:] "... if we confess with our mouth that Jesus is Lord, and believe in our hearts that God raised him from the dead, we will be saved (Romans 10:9). This salvation in Christ is a precious gift and it is nothing that we can earn through our good works (Romans 6:23b 'the free gift of God is eternal life in Christ Jesus our Lord.'; Ephesians 2:8–9 'For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, not a result of works, so that no one may boast.'). The message of the gospel is that we can never be good enough to get to heaven. We must recognize the fact that we are sinners who daily fall short of God’s glory, and we must obey the command to repent of our sins and place our faith and trust in Jesus Christ, who alone was good enough to earn heaven, and who gives that merit to those who believe in His name."
“I encourage you, even today, to repent of your sin and put your faith in Jesus Christ and what He did for you.”
"What is Biblical Repentance and Is It Required For Salvation?" - Proclaiming the Gospel TV with Mike Gendron (WVW TV)
Description: 2 Corinthians 7:9-11 describes Godly sorrow produces repentance unto salvation. Repentance is taught in the Bible over and over. Jesus commanded His disciples to go forth and preach repentance. What exactly is Biblical repentance? Why is repentance so seldom preached, explained, or discussed in modern churches? Why must we discuss and explain repentance when sharing the gospel? What is the connection between the Holy Spirit and repentance? Could that lack of including repentance in the gospel message explain why there are so many false converts? Watch here.
(PDF) Repentance and the Gospel by Mike Gendron
Repenting Unrepentant Repentance - Mike Abendroth - NoCo Radio Part 1
Repenting Unrepentant Repentance - Mike Abendroth - NoCo Radio Part 2