In order to be a living sacrifice, the believer must completely change the way they think about everything. There are two parts to this change of thinking.
First, Paul says the believer is to “not be conformed” to the image of this world. The verb συσχηματίζω refers to being molded into another form, or guided by something else (BDAG). This is a compound word with σχῆμα “The term σχῆμα denotes the outward structure or form that may be known by the senses” (TDNT 7:954-58).
The “pattern of this world” is the way a culture thinks, the Greco-Roman worldview. This would include how a Gentile thinks about the gods, how daily life is regulated by placate the gods, relying on magic or divination when making decisions, etc. The average Roman would think about the Roman empire and the claims made by the emperor quite differently than a Christian view of empires based on the Hebrew Bible. The pursuit of honor in the Roman culture effects how and why a person decides to act in any given situation.
Second, the believer must be “transformed by the renewal of our minds.” The verb μεταμορφόω refer to both outward physical changes (such as the transfiguration, Matthew 17:2) and inward spiritual changes (BDAG). It is used of the change of the physical body in glory (2 Corinthians 3:18). In Romans 12:2 the word refers to an inward spiritual change of the believer by the power of the Spirit. The verb is a passive imperative, suggesting that it is God who does the actual transforming of our minds so that we begin to think differently (Kruse, Romans, 464).
The key to this metamorphosis is the “renewing” of our minds. Paul may have coined the word ἀνακαίνωσις, both the noun and the verb (2 Cor 4:16; Col 3:10) do not appear outside of Christian literature (Jewett, Romans, 733). The word combines the more common καινόω, “to make new” with ἀνα to form a word which means to make something new again, to return it to a pristine state prior to it becoming “unnew.” In Ephesians 4:21-24 Paul describes this process as putting off the old man and putting on the new (Cf. Col 3).
Paul argued in Romans 1 that Gentiles are futile in their thinking and ignorant of the way things really are. But the one who is in Christ has been enlightened, renewed so that they can “think about how they think,” renewing their minds in Christ Jesus.
For example, they would have fully accepted gods had some control over their life, they may have made sacrifices or performed rituals to ensure good luck on a journey, they may have believed people could curse them, or even purchased magical amulets to protect themselves from such curses.
Paul is describing a change in the way we think about everything in life! For example: this new way of thinking includes how people relate to one another. Instead of trying to use people to get ahead in the pursuit of honor and shame, people ought to serve one another in sincerity of love. Instead of seeking revenge, we ought to pray for our enemies.