I realise that posting a chunk of text from a book isn't really much of a discussion, so I wanted to come back and post some of my own thoughts. I think part of what it means to not conform to the world is that we should do good works and be virtuous. The world itself is good, IMO, just like the body is good in itself. But just as the Scripture sometimes speaks of the body in a negative way (especially "the flesh"), so to does it speak of the world in a negative way at times. In these cases it is not the world itself which is bad--for the world is beautiful and a manifestation of the glory of God (Rom. 1:20)--but the fallen elements, that which directs us away from God. But to "not conform to the pattern of this world" means, I think, to be "But be ye doers of the word, and not hearers only," (James 1:22) to be participants in "pure religion" (James 1:27), and to do Christ's commandments, for those who love him follow them (Jn. 14:15, 21; 15:10).
Being Christians, and knowing the Gospel, and knowing all about lofty spiritual things (do we really?), we are held to a different standard. More is expected of us. Jesus said: "If you were blind, you would not be guilty of sin; but now that you claim you can see, your guilt remains." (Jn. 9:41) And James also: "Therefore to him that knoweth to do good, and doeth it not, to him it is sin." (James 4:17) We must therefore do as we know we ought to do. Even those who never heard the Gospel will have to give an accounting (Rom. 2:12-16). To not conform to the world then means to instead conform to the will of God, insofar as we know it or can discern it, as the passage you ask about itself says: "And be not conformed to this world: but be ye transformed by the renewing of your mind, that ye may prove what is that good, and acceptable, and perfect, will of God." (Rom. 12:2)
As for being transformed by the renewing of the mind, this, I think, would be done through the Holy Spirit (cf Tit. 3:5), and involves the cultivation of virtues and having a virtuous spirit, as mentioned in James: "But if ye have bitter envying and strife in your hearts, glory not, and lie not against the truth. This wisdom descendeth not from above, but is earthly, sensual, devilish. For where envying and strife is, there is confusion and every evil work. But the wisdom that is from above is first pure, then peaceable, gentle, and easy to be intreated, full of mercy and good fruits, without partiality, and without hypocrisy. And the fruit of righteousness is sown in peace of them that make peace." (James 3:14-18) The wisdom of this world is foolishness to God (1 Cor. 3:18-20), and the wisdom of God is foolishness to the world (1 Cor. 1:18-25). But we have access to the wisdom of God, by guidance of the Holy Spirit (Jn. 16:13) and having the "mind of Christ" (1 Cor. 2:16; Phil. 2:5). To constantly seek out this wisdom of God, and to cooperate with God as a “worker together with him“ (2 Cor. 6:1; 1 Cor. 3:9), leads to the transformation and renewal that the verse speaks of.
Then it is our job to "shine as lights in the world" (Phil. 2:15; Matt. 5:14-16), and to be the salt for the good of others (Matt. 5:13). Christ's "kingdom is not of this world" (Jn. 18:36), yet we are here, and so "we should live soberly, righteously, and godly, in this present world" (Tit. 2:12), helping others in the way that Paul indicated: "Take heed unto thyself, and unto the doctrine; continue in them: for in doing this thou shalt both save thyself, and them that hear thee." (1 Tim. 4:16) In this way we can follow the advice of Paul to: "present your bodies a living sacrifice, holy, acceptable unto God, which is your reasonable service" (Rom. 12:1).