First, my understanding of the verses was aided when I found out that "renewing" could be thought of as renovation. And that it carries with it the idea of transformation into a new kind of life.
Generally speaking, my mind was renewed by the good news when I realized that I am not my mind, I am not my thoughts. That in me was the kingdom of God. That in me was a whole different mind, the mind of Christ. And that by experiment and by observation, and that by knowing myself, I understood how I was living centered in my natural, carnal mind and proceeding out of that. And I learned how to live centered in this internal "other place", the kingdom of God within me. And I then began living a life centered in the mind of Christ and proceeding out from there. A whole change of center is the key to spiritual life.
Speaking now of the mechanics of renewal, from my new center, I observed the scripts and programmings that run in my natural mind. And when some thoughts pop up that I'd rather not have repeat in me, I stop and let them flow out of me. And I choose again. I choose to hear what the Father has to say about the matter, in me, and then I "eat" the word that proceeds out of the kingdom within me. This plants the seed in me, which grows and bears fruit,