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Prayer is a conversation with Hashem, yes. Part of prayer is the "kavanah," the intention/focus/feeling that you put into it. Even when praying the "kevah" ("fixed") prayers, kavanah is important. As tradition would have it, prayer books were written originally to assist those who, awed by Hashem's might and majesty, were struck dumb and didn't know what to say. So learned people put prayers into words, often using passages from the Tanakh to expound on those things they felt were important: the many ways in which Hashem has saved the Jewish people time and time again)the exodus from Egypt, the Song at the Sea, the binding of Yitzchak); the strength, power, might, and awesomeness of Hashem; the gentle compassion and love of Hashem; the importance of observing Torah law; and yes, the directions for performing sacrifices of animal, meal, oil, and incense.

In a traditional (Orthodox or Conservative) prayer book, these things are expounded explicitly, so that a person who prays those prayers daily in a language he understands will gradually come to know a great deal about Tanakh and about the religious duties that spring from Tanakh, including the sacrifices. Three times a day we pray (shacharit/morning, mincha/afternoon, and arvit/evening), including the Shema (Hear, Israel, and the paragraphs that follow the central tenet of Jewish faith -- wait, actually there's no Shema in the mincha/afternoon service), the Amidah (standing prayer, the central series of Jewish prayers), and the Korbanot (sacrifices). In fact, as part of the prayer, we say explicitly, "May the recitation of these prayers be as if I had offered the sacrifices properly at the Temple," and later we say, "May the words of my lips and the meditations of my heart find favor before you, Hashem, my rock and redeemer." This, I have been told, is because we once said "May these sacrifices and the intent behind them find favor before you."

This is one of the many reasons I highly recommend that a convert should try to convert by Orthodoxy: this isn't taught in Reform, and it's taught but glossed over a little bit by some Conservative teachers. In fact, I highly suggest that all Jews, converted or born, should spend at least a little time learning with an Orthodox or Sephardic congregation, so that these basic points of Jewish knowledge will be learned. That way when a Christian proselytizer comes and says "But we have a way to be redeemed from our sins, and you don't," you'll be able to answer, "Oh, yes we do, as it says in the prayer book, quoting from the prophet in the passage ..."

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How common is this prayer describing sacrifice? How do you pray if you don't mind me asking? Does is suffice to ask G-d for forgiveness and then actually make an effort to change your ways?

We pray the sacrifices at shacharit (morning prayers), mincha (afternoon), arvit (evening; also called ma'ariv which is pronounced mah-ah-REEV, or in Yiddish, mairev and pronounced MY-rev), and again at musaf (the additional prayer said after arvit), in accordance with the traditional (Sephardi, Orthodox, or Conservative) prayer book. Hashem knows that the current generation of Jews aren't neglecting our dutiful sacrifices on purpose -- we are only waiting for the promised mashiach to come, to settle all halachic doubts, and to instigate the rebuilding of our Temple so that we can once again do what Hashem's Torah requires of us. But in the meantime, of course he accepts the prayers from our hearts, willing gifts to charity, and sincere repentance of wrongdoings. In fact, the Talmud tells us this, and we recite it every Rosh Hashanah and Yom Kippur, if not more often:

"On Rosh Hashanah it is written; on Yom Kippur it is sealed. But prayer, charity, and repentance will avert the severest decree!"

What is written? Our fates for the coming year.
What is sealed? Our fates for the coming year.
Why ten days between the writing and the sealing? To give us time to pray for forgiveness; to ask our neighbors to forgive us for our wrongdoings, and to forgive them as well; to give a material sacrifice to help those in need, to show a turning of our hearts; to repent truly from what we have thought and felt, which made us act wrongly, and to show Hashem that if similar circumstances arise before us, we will take the better path next time.

Ah, yes, thanks. It was on the tip of my tongue, Shavuot.