The Christian Perspective
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Below are the verses cited in the question. John 3:10 doesn't seem to fit the question, so I assume it was a typo. Nonetheless, the other verses should more than suffice to make the point the questioner intended.

Genesis 6:7
The LORD said, "I will blot out man whom I have created from the face of the land, from man to animals to creeping things and to birds of the sky; for I am sorry that I have made them.

Exodus 32:14
So the LORD changed His mind about the harm which He said He would do to His people.

Judges 2:18
When the LORD raised up judges for them, the LORD was with the judge and delivered them from the hand of their enemies all the days of the judge; for the LORD was moved to pity by their groaning because of those who oppressed and afflicted them.

1 Samuel 15:35
Samuel did not see Saul again until the day of his death; for Samuel grieved over Saul. And the LORD regretted that He had made Saul king over Israel.

Some of these passages seem to depict a God who has no clue what He's doing. He makes mankind and then decides, whoa, that was a big mistake. So He floods the world and kills everyone. He makes Saul king and then realizes that was not the best idea He ever had, etc.

Of course, we know that this is not the case. The Bible tells us so.

For example, with respect to Saul, before making him king, God announced clearly to the people of Israel that their idea to crown a king immediately was not something they would want if they knew better (1 Samuel 8:4-22). So we can see that God was fully aware of the problems beforehand, and knew exactly what He was doing when He anointed Saul as King.

In fact, God knows the future perfectly (Isaiah 42:9, 44:7, 48:5). After all, He is the one who makes it what it is (Acts 4:28; Isaiah 10:15; Romans 9).

Since we know this, we know that God does not actually change His mind, or regret His own actions, etc. In fact, God states very explicitly, "I, the LORD, do not change" (Malachi 3:6).

So how are we to understand the verses that claim that God changes His mind?

In Genesis 6:7, the word translated "sorry" can also mean "grieved". Taken as such, it is easy to understand it as an anthropomorphism, or in other words, God describing His divine thought processes to us using human characteristics in a way that we can relate to. There is no real way that humans can fully comprehend the mind of God. Anthropomorphisms help to put Him into a more human perspective.

We have been grieved before, we have been upset before, and when we are, we do what needs to be done to change the situation. This is essentially what God did. It was always His plan to do these things, but the anthropomorphism helps us to understand why He did it, even though, like all anthropomorphisms, it is metaphorical and thus should not be taken literally.

In Exodus 32:14, God made a threat apparently in order to strengthen Moses' faith. The verse is simply informing us, again anthropomorphically, that God did not have to carry out His threat because Moses' response was satisfactory.

In Judges 2:8, God responded, as He had always planned to do, to the groaning of His people, which He knew would come at that exact time.

In 1 Samuel 5:35, as we have already shown, God clearly knew in advance what was coming. Nonetheless, a just God punishes sin. It was time to punish Saul, and the verse in question, along with its anthropomorphic "grief" (the same Hebrew word as in Genesis 6:7) simply serves to show us how reprehensible our sin is before God.

God is our Creator, but He also wants us to know that He will adopt us as sons if we trust in Jesus Christ for the forgiveness of our sins (Ephesians 1:5). He is transcendent (beyond the universe) but also immanent (within the universe), personal (having a volitional will), and relational (able to connect directly with us).

Anthropomorphisms serve to show us God's immanence, personal qualities, and relational nature, in spite of His incredible transcendence. They are used frequently throughout the Bible to remind us that, in addition to being a timeless Creator and sinless Judge, God can also be a friend who sticks closer than a brother (Proverbs 18:24).

Amos 4:13
For behold, He who forms mountains and creates the wind And declares to man what are His thoughts, He who makes dawn into darkness And treads on the high places of the earth, The LORD God of hosts is His name.