To consecrate oneself to the Sacred Heart means to make all of your actions, thoughts, desires to be for the Sacred Heart. It means you strive to become more like the Sacred Heart - enflamed with a true, sacrificial love for the world. The Sacred Heart, remember, is a display of how real the love of God for us is: it is so real, He could not but help to take on our physical flesh, and not only that, but to die so we might be free from bondage. It is a devotion which calls its devotees to ponder constantly that most mysterious and incomprehensible love.
The Sacred Heart of Jesus and Immaculate Heart of Mary go hand-in-hand, really. There is a profound theological meaning to being devoted to the Two Hearts. If the Sacred Heart is the goal, than the Immaculate Heart is the means: to imitate Mary Immaculate means to be humbly pure from sin - and that is the only way we can open ourselves truly to be set on fire with the love of God.
EWTN’s page is a good one, although I think I have found two other forms of the prayer. Mostly translation differences. I would strongly recommend you become consecrated to our Lord’s Most Sacred Heart, and, if you do, to also discern whether or not to become consecrated as well to Mary’s Immaculate Heart.
I myself am a convert to the faith - and Jesus’ Sacred Heart too drew me. I was consecrated this morning to His Sacred Heart, after Mass. Welcome home!