It has been said that a Christian has a mind through which Christ thinks, a heart through which He loves, a voice through which He speaks, and hands through which He helps--this is the epitome of spirituality--to know Christ and make Him known.
"O that they were wise, that they would understand this, that they would consider their latter end!" (Deut. 32:29, KJV).
That was the title of the 1918 book by Lewis Sperry Chafer, the founder of Dallas Theological Seminary, that made him a renowned and celebrated theologian. Who is? This is a vital and bona fide question: Like G. K. Chesterton has said, "We have found all the questions, now let's find the answers!" When we are spiritual, we exhibit the fruit of the Spirit in a manifold manner. There is no certain manifestation, such as talking about Jesus or the Bible. Sometimes just touching base with someone in love and charity and meeting their needs is genuine fellowship and an expression of being spiritual.
Just think of all the possibilities of expressing the nine winsome graces given by the filling of the Holy Spirit. Wherever two or three are gathered together in Jesus' name, there He is. The spiritual one simply walks in the Spirit and has continual fellowship with the Lord (keeping short accounts of his sins and confessing them per 1 John 1:9: "If we confess our sins, He is faithful to forgive us our sins, and to cleanse us from all unrighteousness."
No one can claim to be always spiritual or that they have "arrived" at such a point of perfection, of not being conscious of sin or shortcomings. Sometimes the wisest remarks can proceed out of the mouths of infants (cf. Matt. 21:16), as Jesus noticed: Psalm 8:2 says, "Through the praise of children and infants..." I believe God can even use children: a child's voice convicted St. Augustine: "Take and read, take and read." Proverbs 20:9, HCSB, says, "'Who can say, "I have kept my heart pure; I am cleansed from my sin?'"
He that is spiritual simply walks with the Lord as Enoch and Noah ("Noah was a just man and perfect in his generations, and Noah walked with God," Gen. 6:9)--and we have this privilege too! It is a "faith-walk" because "we walk by faith, and not by sight" (2 Cor. 5:7). There is no veneer to see through or guise of spirituality, such as hypocrisy (he has nothing to hide, and is straightforward in speech), but a genuineness and authenticity in action. He is the real thing, an original! He's not out to outshine someone or be a rival. "The person with the Spirit makes judgments about all things, but such a person is not subject to merely human judgments' (1 Cor. 2:15). There is a certain natural ability to discern the Spirit, in other words. Whatever he does, he does to the glory of God (cf. 1 Cor. 10:31)!
There is no inherent dichotomy or division of believers into classes of spiritual and non-spiritual, first-class and second-class, or what Chafer mistakenly believed to be carnal and spiritual Christians. Just like it is wrong to have a "holier than thou" attitude (cf. Isa. 65:5), it is incorrect to deceive yourself into thinking you are more spiritual than your brethren--you either are spiritual or you're not--there are no degrees to graduate to.
This doctrine need not be problematic or an issue at all: "So I say, walk by the Spirit and you shall not gratify the desires of the flesh" (Gal. 5:16). We are indeed free in Christ: not free to live according to the flesh and our old nature, but power to live in the new nature or spirit. The old nature knows no law, the new nature needs no law! In other words: Freedom to do what we ought, not what we want! We've never had the right to do what is right in our own eyes or to do what is scripturally wrong. In sum, "So we live by the Spirit, let us keep in step [pace] with the Spirit" (Gal. 5:25). Soli Deo Gloria!