My first driving lesson

Today, I  had my first driving lesson. After a ten minute explainaion from my teacher on what I needed to do, I began to drive. It was my first time and it was incredible! 

In the begining, I had to drive to an area where there were just a few people. There I learned to start the car. Once I was able to control the car, the teacher sent me to a more populated area. At first I was really nervous and I drove very badly. But, after a little practice, I was able to control the car much better.

At the end of the class, the teacher told me to stop the car on a ramp. Then, I had to start again but the car wouldn't roll backwards. It was really dificult.

In the end, nothing has be damaged and I think I have learn a lot of things as a first time car driver.

My grammer is not as good as it should be. But, I do hope this with the other "corrections" will be helpful to you.

My first driving class

Today I have had (simple past tense)

OR

Today I have had (present perfect - Today is wrong in this case)

It's too hard if you don't use simple past tense!  I give up the attempt.

...my first driving class. lesson. After ten minutes (the teacher had been explaining to me what I had to do) I have started to driving. It was my first time and it was incredible.
At the begining, first I drove in have been driving by an area where there were few people. There I have learned to start the car. In the moment in which Once I could controled the car correctly, the teacher has sent took me (he is still with you, so take, not sent) to another area with where there were many people. I was really nervous and in that moment I drove was driving very badly, (needs the adverb) but in few after a short time I was able to control the car better.
In At the end of the lesson class, the teacher has told me to stop the car in on a ramp. In that situation, I had to start again the car again without letting the car roll back but the car didn't have to fell down. It has been was (past simple tense again) really difficult.
At finally, In the end nothing has be had been (past perfect) damaged (or 'no damage had been done') and I think I have learned/learnt lots of things.

This has been/was (this is correct use of present perfect but it is easier to use simple past) my first lesson class

(class is not exactly incorrect - but it refers to 'a group of students' taking lessons.  For one person I prefer lesson)

My first driving class

Here you are mixing tenses, and it makes it difficult to revise - you are alternating between 'has been' and was - let me highlight them first; then we can pick one or the other.

Today, I have had my first driving class. After ten minutes (the teacher has been explaining me what I had to do) I have started to drive. It was my first time and it was incredible.
At the begining, I have been driving by an area where there were few people. There I have learned to start the car. In the moment in which I controled the car correctly the teacher has sent me to another area with many people. I was really nervous and in that moment I drove very bad. But in few time I was able to control the car.
In the end of the class, the teacher has told me to stop the car in a ramp. In that situation, I had to start again the car but the car didn't have to fell down. It has been really dificult.
At finally, nothing has be damaged and I think I have learn lots of things. This has been my first class

My first driving class

Today, I have had my first driving class. The teacher started by explaining to me what I had to do.  After ten minutes (the teacher has been explaining me what I had to do) I have started to drive. It was my first time and it was incredible.
At the begining, I drove by an area where there were few people. There, I have learned to start the car. In the moment in which I controled the car correctly the teacher has sent me to another area with many people. I was really nervous and in that moment I drove very badly. But in short time I was able to control the car.
In the end of the class, the teacher has told me to stop the car in a ramp. In that situation, I had to start again the car but the car couldn't roll backward. It was really dificult.
Finally, nothing was damaged and I think I have learned lots of things. This was my first class.

[Good work, but your story was in the past and so should be in the simple past tense.]