For your writing, I would make the following suggestions: first, proofread very carefully for easy-to-miss mistakes. One such error is found in the first sentence with the homonyms, which are...

...one of a group of words that share the same spelling and the same pronunciation but have different meanings.

"Knew" is a form of "knowing," while "new" refers to something that has not been used—or experienced—before. You want to use "new."

Another aspect of your writing that should be changed is the use of different "tenses," which...

...places an action or situation in the past of the current moment...

The "present tense" is the present moment ("now"). E.g., you note, "As we settled," which is the past tense. In the next sentence, however, you switch to the present tense: "the moist air tingles" and "I breathe." If you follow the tense presented in the first sentence with the verb "settled," then the next sentence (and subsequent sentences) should reflect the past tense—using "tingled" and "breathed."

Next, don't "zoom into the highway," but "zoom onto the highway."

In the following sentence, the simile used is very nice, but I would consider a word different than "levels."

The roads were as smooth as a bowling-green, and we swung along over the levels, up hills and down hills. 

It is easy to find uses of figurative language in your writing: some of the imagery is quite nice. However, there are some aspects of this piece that I would clarify. For instance, the atmosphere cannot be seen, therefore I don't see how it can be "transparently pure." You can say it is pure, but "transparently" infers a state of being seen through, where the atmosphere can never be seen.

At the same time, I don't see how the trees can "glitter." If they were wet with rain, the water might make them seem to glitter if the sun came out—then this would be very effective imagery. I also find the "dazzling heat" poetic but inaccurate...for the heat is not something we see, but something we feel. Light is dazzling, or diamonds. I don't feel that "dazzling" makes sense here. 

The beginning of the second paragraph loses some of its cohesiveness. As mentioned earlier, the first and second sentences are written in different tenses. "The gigantic scale of everything strokes me" does not make sense to me. The idea of "freedom" in the first sentence is not related to the "gigantic scale" you mention; additionally, "strokes" is not used effectively in this context. I would change the next sentence to read "I was looking at..." I would follow it with "Nothing was the same," which is less awkward.

The last several sentence need some tweaking as well. Don't mix your tenses, and pay attention to the form of the verb phrase "to have" that you use. You would want to mention that you new "some" time "would have to pass" before you "had" a "sense of belonging." In the next sentence, "have" must be changed to "had" again.

The last two sentences are fragments: they need a verb, or need to be connected to the previous sentence. You can write, "It is a place where..." However, "future dreams succeeding" is awkward. You could write "dreams of the future coming to pass" (which means "taking place"). The last sentence would also need "It is..." added. You can also join the sentences—as a continuation of the sentence before (so that the last pair of sentences wouldn't be fragments) would be to use a colon after "Australia." So it would read "Australia: a place..."

How can I improve this memoir?

As we settled in the car and put on our seat belts, which was something new to me, I turned down the window. The moist air tingled against my skin, and I breathed it in. Warm wind hit my face as we zoomed onto the highway. The roads were smooth and lined so neatly. Car lanes were organised; cars stopped when they had to, drove when it was their turn. No one seemed to break the rules; no one was pressing the horn every two seconds, no one was careless. The roads were as smooth as a bowling-green, and we swung along over the roads, up hills and down hills. The sky was the most perfect azure, without a cloud, and an atmosphere so pure. The trees wavered in the wind, leafs fluttered to the ground and blooming flowers danced in the slight breeze.

I felt as free as the wind blows, as free as the grass grows. I was looking at a world so different compared to the one I grew up in. Nothing was the same. I knew that some time would have to pass before I had a sense of belonging, but a part of me had already fallen in love with this country, Australia: a place where I can see my dreams of the future coming to pass. It is a place where danger won’t be seen in every corner I turn to.

The essay is a good start to a personal experience or reflection assignment. The main issue that I think needs work is that it seems oddly detached from actual time or place. Although you've made a good effort to use descriptive metaphors, I actually have little sense of specific place. To localize your essay, instead of talking about some generic car journey, can you think of one specific one. Where did it start? What time was it? To give a sense of a real experience, you might include a list of people in the car and where you were sitting (front seat? back?). What is the name of the road you traveled on? What model car was it? Always try to use specific rather than generic terms, i.e. rather than "trees" put the actual species.

How can I improve this memoir?

As we settled in my Dad’s car and put on our seat belts, which was something new to me, I turned down the backseat window. The moist air tingled against my skin, and I breathed it in. Warm wind hit my face as we zoomed onto the highway. Car lanes were organised; cars stopped when they had to, drove when it was their turn. No one seemed to break the rules; no one was pressing the horn every two seconds, no one was careless. The roads were as smooth as a bowling-green, and we swung along over the roads, up hills and down hills. The sky was the most perfect azure, without a cloud, and an atmosphere so pure. The trees wavered in the wind, leafs fluttered to the ground and flourishing flowers danced in the slight breeze of the scorching afternoon. Oh, this country is a paradise utterly beautiful my dreams wouldn’t picture such world.

I felt as free as the wind blows, as free as the grass grows. I was looking at a world so different compared to the one I grew up in. Nothing was the same. I knew that some time would have to pass before I had a sense of belonging, but a part of me had already fallen in love with this country, Australia: a place where I can see my dreams of the future coming to pass. It is a place where danger won’t be seen in every corner I turn to.

I get the contrast you are drawing and this makes sense.  If I were to offer up a suggestion, I believe that it would reside in bringing out the lush greenery of Australia with the arid conditions of Iraq.  I think that there is a way to amplify both in the conditions of one another.  If you wanted to make one superior to another, you could do so.  Yet, I think that in contrasting the desert winds, the sand, and the heat of Iraq with the lushness you experience in Australia, both come alive in the mind of the reader.  This would also fit the idea that the person who is living in one setting over another is in a type of "dual consciousness" where the mind is constantly going back to one over another.  I think that this would be good to bring out because it helps to illuminate the complex nature of the individual who is transplanted from one place to another and how consciousness is divided between where we are and where we have been.  In bringing out the contrasts in both climates, the memoir moves closer to accomplishing this divided state of the individual and helps to establish the complexity in both the narrative and its effect on the reader.