To flee means to avoid at all costs by having a predetermined mindset that you will not participate in any way with things that offend your Heavenly Father.

What are youthful lusts? Primarily illicit sexual desires but within the context of 2 Tim. 2 it includes such lusts as pride, desire (passion) for wealth, a desire for power that is to say authority over others, jealousy, self-assertiveness, and an argumentative spirit. Verse 16 includes these profane (words) and idle babblings, meaning the use of human reason to subvert God's word. This is extremely dangerous. Why avoid these things? Because they lead to V. 14 - ruin, V. 16 - ungodliness (behavior that is not pleasing to God as written in the Bible), V. 17 - it will spread to other areas of your life like a cancer, V. 18 - it will take you further and further from the truth of scripture.

Now how do believers avoid the enemies trap? Look at verse 15 - diligently study the word of God which is absolute truth that leads to godly behavior and discernment. V. 16 - shun all ungodly behavior and conversations, which means to turn one's self about for the purpose of avoiding something all together. This must be predetermined in your heart. Listen to David's confession -
1 I will sing of mercy and justice;
To You, O LORD, I will sing praises.
2 I will behave wisely in a perfect way.
Oh, when will You come to me?
I will walk within my house with a perfect heart.
3 I will set nothing wicked before my eyes;
I hate the work of those who fall away;
It shall not cling to me.
4 A perverse heart shall depart from me;
I will not know wickedness. Psalms 101:1-4

Now look at verse 19 - "Let everyone who names the name of Christ depart from iniquity." 2 Timothy 2:19 We ought to pursue righteousness, that which is in accordance with scripture with reckless abandonment.

Now the Holy Spirit provides additional instructions on how the believer is to conform to this pattern of behavior. It is given to us in verses 22 - 25:

This is the spiritual side:
1. Pursue righteousness - what's that? integrity, virtue, purity of life, rightness, correctness of thinking, feeling, and acting
2. Pursue faith - this is faithfulness of behavior towards the things of God, meaning to trust that as you commit yourself to obeying the word of God that He will provide guidance, discernment, and strength.
3. Pursue love - this is the love of God's truth, that is the word of God. Fall in love with God's word, read it daily requesting wisdom to apply the knowledge obtained through understanding of the scriptures.

This is the practical side:
1. Avoid disputes - don't participate in conversations that are outside of scripture. Give Biblical references but once it turns to a conversation about vain things then cease from commenting further. Why? Because it leads to strife or arguments which are useless. Remember the context is concerning "false teachers" not the everyday skeptic.
2. Don't quarrel - what's that? Those who engage in a war of words, wrangling in disputes. How do you avoid that?
i. By being gentle in tone and attitude,
ii. Being able teach or instruct by using the scriptures not your opinions,
iii. By being patient, allowing the person to fully explain their position then interacting with them from the scriptures not your opinions,
iv. Speaking with humility, not making the person feel like they are less than you or a greater sinner than you because they are not. I am the greatest sinner I know because I know me better than anyone else besides the Lord.

What's the purpose for all this godly behavior, knowing the scriptures, being gentle, humble, etc...? The Holy Spirit tells us in verse 25,
1. Peradventure that God will grant them repentance - to recognize their sinful condition when compared to a holy righteous God, not to us.
2. That they may know the truth - what truth? The record that God gave of His Son, Jesus Christ, the gospel. Christ died to save sinners, was buried to take our separation from God, was raised from the dead because God was satisfied with His payment (death) for our sins. This is what must be believed and accepted. My sins caused my Savior His life, that His life may give me eternal life.
3. That they may come to their senses - this is repentance
4. Escape the snare of the devil - this is unbelief
5. To be free from captivity by the devil to do his will - this is salvation. The believer goes from being a slave to sin and the devil to being a slave to righteousness and God.

There you have it my friend, the what, the why, and the how.

Hope that helps.