We rented a car, whatich was really unexpected, because we did not have international driver’s permits and no one of us have an experience with left-hand traffic.

Usually "experience" is uncountable in English, unless you're referring to specific instances that happened.

That day we were driving along the road from York to northward, when we stopped to drink coffee and eat somea meal.

"Some food" or "a meal". Food is uncountable here, and meal is countable, so you would use "some" if you say food, or "a" if you use meal.

I started looking at them and I was surprised when I saw the photo on which a sycamore in a gap was depicted.

"Depicted" sounds a bit out-of-place if it's a photograph. A painting depicts something, but a photograph really just shows it exactly how it is. It might just be me though.

It was themy first time when I visiteding England, but I was absolutely sure I knew that tree.

You might choose "recognized" here instead of "knew" for a bit more clarity.

In my childhood I watched movie that is called ‘Robin Hood: Prince of Thieves’ maybe ten or more times.

I really liked the plot, roles and costumes and especially the nature of England.

This is fine but maybe consider using more commas, especially Oxford commas between the second-to-last and last items in a list. It's mostly just my preference as a Canadian though.

So at the moment when I saw that picture I decided that I had to visit that location.

You shouldn't start a new sentence with "so", or any conjunction. Just join it to the previous sentence with a comma.

InTo my delight, it was nearby us.

What you wrote is clear but sounds very unusual, probably just because "to my delight" is a common set-phrase. You generally don't use an object with "nearby", as it's implied to be the subject (more more strongly than "near").

In forty minutes we were walking along the Hadrian’s Wall.

I think it's because I wore a green knitted sweater with a pointed hood on that particular day.

This is a bit incomplete for an English sentence. It's lacking a clear subject/verb, and ends up sounding like a fragment. You need to use this dummy subject/verb "it is" in these cases to make it grammatical.

Eventually, we found that amazing sycamore, and I took couple memorable photos and then I hugged it goodbye as if it were my good friend from my childhood.

This is nice, but a bit more parallelism between the phrases would help it sound better. You have three past-tense verbs (found, took, hugged) which is perfect, but the subjects hop around a bit (we, I (ommitted), I). It would probably be fine if you just insert the "I" implied in the second phrase.

MA month ago, my mother presentedgave me a book is called ‘The Great Atlas of Wonders of the World’.

"Presented" needs some sort of preposition. "Presented a book to me", not "Presented me a book".

I remembered that book and that time when I was a little girl and was not be able to read books.

H; however, I was a big fan of it and I flipped through the book, admired the pictures and imagined which places I would visit when grow up.

Shouldn't start sentences with "however". It technically requires a semicolon to join with the previous sentence, even though that's a bit uncommon in modern English.

So, of course, I opened my gift and I was stunned.

You've started another sentence with "so" which is not proper for written English, but this is very natural for spoken.

Since that moment I've started to think about dreams.

What is the difference between our childhood dreams and our adult ones?

Your way is fine, but a second "our" sets up the comparison more clearly.

It is a measurable and conscious activity.

Your way is okay but without "a" it sounds very abstractly philosophical.

However, when I was a little girl I would just focus on the thingsa goal without thinking about how I will beould achievinge it.

There are two different ways for beingto be happy and both work, (am I right?).

Comma splice at the end, since both are independent clauses.