Adult themes should start with being right with God judicially, then walking with God practically: dealing with sin, learning a skill and getting and keeping a job, doing good work, being truthful with no lying, keeping commitments, getting and staying married, being a God-fearing spouse and parent, being involved in serving the Lord in church.
Adulthood as maturity is sketched out memorably in a couple of Biblical passages. Let's start with Ephesians 4:11-16 —
And he gave the apostles, the prophets, the evangelists, the shepherds and teachers, 12 to equip the saints for the work of ministry, for building up the body of Christ, 13 until we all attain to the unity of the faith and of the knowledge of the Son of God, to mature manhood, to the measure of the stature of the fullness of Christ, 14 so that we may no longer be children, tossed to and fro by the waves and carried about by every wind of doctrine, by human cunning, by craftiness in deceitful schemes. 15 Rather, speaking the truth in love, we are to grow up in every way into him who is the head, into Christ, 16 from whom the whole body, joined and held together by every joint with which it is equipped, when each part is working properly, makes the body grow so that it builds itself up in love.Here it is a matter of spiritual maturity, specifically of doctrinal maturity, ability to stand firm and not be rocked around by every passing fad.
In this context, it is interesting that the apostle goes on to mention what we are calling "adult themes" — that Christians should abstain from sensuality and impurity (v. 19).
When the writer to the Hebrews mentions "adult themes," he is pretty forceful about it:
About [Melchizedek] we have much to say, and it is hard to explain, since you have become dull of hearing. 12 For though by this time you ought to be teachers, you need someone to teach you again the basic principles of the oracles of God. You need milk, not solid food, 13 for everyone who lives on milk is unskilled in the word of righteousness, since he is a child. 14 But solid food is for the mature, for those who have their powers of discernment trained by constant practice to distinguish good from evil. (5:11-14)Again, the focus is on doctrinal maturity which shows itself in moral/ethical maturity, being able to distinguish good from evil.
Further, the verses that follow show that the writer saw maturity as far more than a desirable extra. His view was: grow, or crash. Their failure to grow actually made him fear for the reality of their conversion.
Obviously, the apostles' view was far different from ours. The concept of baby men was unthinkable. The new life implanted by God through His word naturally impels us to grow, strive, mature. Not always at the same rate; some bear 100, but also some 60 and some 30 (Matthew 13:23) — but all bear fruit.
So next time you read or hear this abuse of "adult themes," to mean "sniggery juvenile themes," roll your eyes, shake your head, and redouble your prayer and commitment to grow up — which means growing out of such things.