We needed to stpay alert about all our things, and always being withttention to our stuff and keep our eyes openeled.

This sounds more natural

There, something curiousodd happened.

“Curious” isn’t really used to describe objects or concepts, it’s more used for people and animals. The word “odd” would be better here

I didn't know then, but many owners of these shops awere from Asia.

So, while we were shopping, my mother got close to a guy, he waswalked up to the cashier, and he was so stressed because he didn't know Spanish.

1. “To get close to someone” is a phrase used more for emotional closeness. It works in physical closeness, but the way you used it feels like you were trying to say she started getting to know the guy
2. Additionally, saying “she got close to a guy, he was the cashier” is a bit lengthy and can easily be cut down by combing these two parts of the sentence

It was funny, he tried to say a few words but finally he surrendered in the middle ofgave up, laughs.ing.

1. “Gave up” sounds better than “surrendered” in this context
2. “In the middle of laughs” sounds weird, and I can’t quite tell what you mean. Did you mean he was laughing?

We needed to stay alert about all our things, and always being withkeep our eyes open.

Theren something curious happened.

This type of sentence is only for time / sequence. It was there something curious happened is the normal sentence that can be for space.

I didn't know, but many owners of these shops' owners are from Asia.

many owners of these is okay but disturbs the phrase 'many of _', so am not sure about it. shops' is also a bit weird but it's a corner. if you said 'I didn't know while there,' you could just say owners, for example.

So, while we were shopping, my mother got close to a guy, h. He was the cashier and he was so stressed because he didn't know Spanish.

While is the preposition of a prepositional phrase, so can't go in an parenthetical clause. So is just emphasis, you don't need to emphasize it more with a comma at the expense of coherence. Also this is fine to be two sentances, which also re-highlights the so a bit more. And would need to be a coordinating conjunction ie have a comma if both independent clauses start with 'he', by removing one he you can accurately get rid of some repetition and make one clause dependent, so you don't need the comma.

It was funny, he tried to say a few words but finally he surrendered in the middleto a fit of laughster.

Same thing here, and 'fit of laughter' 'laughing fit' is a phrase and is one of the meanings of 'fit'.

Shopping with mMy Mom

Today, my mom and I went shopping.

I was a bit stressed, because it was in a dangerous neighborhood here in my city.

We needed to stay alert about all our things, and always being withhaving our eyes open.

Theren, something curious happened.

I didn't know it before, but many of the owners of these shops are from Asia.

So, while we were shopping, my mother got close to a guy, he wasfriendly with the cashier and h. He was so stressed because he didn't know Spanish.

It was funny, h. He tried to say a few words, but finally he surrendered in the middle of laughs.

My mother is so friendly, that she can makes friends in any place.