I’ve started a new sport this year, and I’m happier than I could expected!

Add a comma before “and.”
Use the past tense “expected” to match the time frame.

I learnted to swim at school when I was about 10 years old, but my technique was pretty basic, and I was not confident at all.

“Learned” is the standard American English form (though “learnt” is fine in British English).

For example, I didn’t put my head under the water, as I didn’t like it, but you can’t have answim efficient swim bly this way, and you will be quickly become exhausted.

Swim efficiently” is the natural phrasing.
Add a comma for clarity.
“Quickly become exhausted” is smoother.

On lLast September, I moved into a new house, and by chance, there is a public swimming pool atwithin less than ten minutes by foot.

Remove “On.”
Use “moved into” instead of “moved in” when referring to a house.
Change preposition.

I subscribedigned up for the lessons for a year, one hour a week, except during the school holidays when there are no lessons.

“Signed up for lessons” is the natural phrasing.
Put a comma before "except"

Thanks to the coach’s exercicses and support, I haven’t given up, and I’ve even made such progress thant I’m now able to do a swim lap — 25 metres — almost without a pause for 3 swimming styles — front crawl, breaststroke and backstroke.

Spelling correction.
“Such progress that” is the correct phrase.

I’ve started a new sport this year and I’m happier than I could expect!

For example, I didn’t put my head under the water, as I didn’t like it, but you can’t have an efficient swim by this way and you will be quickly exhausted.

On last September, I moved into a new house and by chance there is a public swimming pool at less than ten minutes by foot.

As soon as I lost my concentration, the punishment was immediate :— I swallowed water, I choked, and I panicked!

Thanks to the coach’s exercices and support, I haven’t given up and I’ve even made such progress than. I’m now able to do a swim lap —of 25 metres —, almost without a pause, for 3 swimming styles — front crawl, breaststroke and backstroke.

I’ve started a new sport this year and I’m happier than I could expectever!

For example, I didn’t put my head under the water, as I didn’t like it, but you can’t have answim efficient swim by this way andly like that so you will be quickly be exhausted.

On lLast September, I moved into a new house and by chance there iwas a public swimming pool at less than a ten minutes by foot walk away.

So, no more excuse anymores!

I subscribed theigned up for lessons for a year, one hour a week except during the school holidays when there are no lessons.

As soon as I lost my concentration, the punishment was immediate : I swallowed water, I choked and I panicked!

A comma can be used here instead of a colon, but I think both work fine.

Thanks to the coach’s exercicses and support, I haven’t given up and I’ve even made so much progress than I’m now able to do a swim lap — 25 metres — almost without a pauseresting for 3 swimming styles — front crawl, breaststroke and backstroke.