Yes, it is true the belief you were raised with is not quite biblical. That is, we should not be afraid of Satan. When we sin, we shouldn't expect that Satan would go attack our family. We are primarily responsible for our own sins.
It is clear from the Old Testament itself that we are responsible for our own sins:
"The son will not bear the punishment for the father’s iniquity, nor will the father bear the punishment for the son’s iniquity; the righteousness of the righteous will be upon himself, and the wickedness of the wicked will be upon himself." (Ezekiel 18:20)
So it is not true that if you sin, someone else in your family will suffer instead of you.
While a person's sin can affect others (e.g. in the case of Adam and Eve and Achan), suffering can happen for various reasons. As in the case of Job, it was not Job's fault that calamity fell upon him and his family. In fact, when Job maintained this point and did not curse God, he received blessings from God.
One of the reasons Job suffered was to show this point early on to many generations of believers through the Bible that suffering is not necessarily linked to sin.
And Jesus reiterated this point when His disciples saw a man born blind.
"Jesus answered, 'It was neither that this man sinned, nor his parents; but it was so that the works of God might be displayed in him.'" (John 9:3)
So the blind man's suffering was not connected to his sin or his parents' sin. But God allowed him to be blind so that Jesus' miracle can come upon him. Not only was this man healed, but his story lives on to teach many generations after him.
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There are four general reasons why Christians suffer.
1. We suffer because we live in a fallen world.
In a fallen world, where our own bodies are subject to corruption, we could get easily sick and hurt. Others can also do harm to us intentionally or unintentionally.
2. We suffer because we are being disciplined by God.
God disciplines us when we sin as Christians, so that we do not become spoiled children. One example of this is when the Corinthian church took communion without respecting Jesus. Many of them treated communion in an unworthy manner.
So it was written:
"Therefore whoever eats the bread or drinks the cup of the Lord in an unworthy manner, shall be guilty of the body and the blood of the Lord. But a man must examine himself, and in so doing he is to eat of the bread and drink of the cup. For he who eats and drinks, eats and drinks judgment to himself if he does not judge the body rightly. For this reason many among you are weak and sick, and a number sleep. But if we judged ourselves rightly, we would not be judged. But when we are judged, we are disciplined by the Lord so that we will not be condemned along with the world." (1 Corinthians 11:27-32)
This passage shows God disciplines us when we sin to keep us from sinning further.
3. We suffer to help us grow and to help others.
Sometimes we may not know the reason we suffer. But when we do suffer, we tend to seek God more and grow more through our sufferings.
The Bible says:
"And not only this, but we also exult in our tribulations, knowing that tribulation brings about perseverance; and perseverance, proven character; and proven character, hope; and hope does not disappoint..." (Romans 5:3-5)
For example, Timothy suffered stomach pains that could not be healed by any of the apostles.
So Paul wrote:
"No longer drink water exclusively, but use a little wine for the sake of your stomach and your frequent ailments." (1 Timothy 5:23)
Timothy was probably very surprised that his sickness could not be healed miraculously. But we can be sure he grew through his sufferings. And we know that his suffering was for the benefit of us all, because this passage shows God can heal both through medicine and miracles.
The wine acted as an antiseptic to help cleanse the water Timothy was drinking. For anyone who refuses to take medicine, we can use this verse to encourage them.
4. We suffer because we are persecuted for our faith.
Jesus told us that we should rejoice when others persecute us, because great is our reward in heaven (Matthew 5:11-12).
Paul suffered greatly because of the gospel, but he is greatly honored and rewarded.
So there are many reasons for suffering and it is not just Satan trying to get us. Sometimes God allows Satan to attack us, but God is always in control.
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Finally, we need to treat sin seriously, not because of fear of Satan getting us or our family. But we should fear God, so that we refrain from purposefully sinning.
Jesus said: "Do not fear those who kill the body but are unable to kill the soul; but rather fear Him who is able to destroy both soul and body in hell." (Matthew 10:28)
We should fear God who is the great Judge so that we do not sin casually.
The Bible says: “For if we go on sinning willfully after receiving the knowledge of the truth, there no longer remains a sacrifice for sins, but a terrifying expectation of judgment and the fury of a fire which will consume the adversaries” (Hebrews 10:26-27).
If we continue to sin casually, then we haven't truly believed. Because if we truly believe in Jesus, then we would do what He says. Believing in Jesus is having faith in Him and learning to be faithful too.
But if we have any sins, ask God's forgiveness and repent. "He is faithful and righteous to forgive us our sins and to cleanse us from all unrighteousness" (1 John 1:8-9).
Repent of any sins and learn to follow Jesus diligently. Rely on God and work with Him by prayer and diligence to overcome any known sins.
May God bless you.