Can you please check these few sentences? Thank you.

1)I insert a 50-cent coin into the trolley and I do my shopping.
2)I need to buy 200 grammes cooked ham and 150 grammes of raw ham.
3)Then I need to go to the counter of sliced meats and salami and then to that of fruits and vetegables.
In the winter I go bobsledding and sledging with my brother.
4) I always put everything I need in a trolley or a plastic basket.
How do you call the shopping card?

I can tell you have a British teacher. :-)

"1)I insert a 50-cent coin into the trolley and I do my shopping."

Ok


"2)I need to buy 200 grammes cooked ham and 150 grammes of raw ham."

200 grammes OF cooked ham

"3)Then I need to go to the counter of sliced meats and salami and then to that of fruits and vetegables."

I assume this means to say
"Then I need to go to the sliced meats counter and buy salami and then go to the fruits and vegetables counter." There usually wouldn't be a counter for that, but there would be nothing gramatically wrong with that. It would sound better to make it 2 sentences.

"In the winter I go bobsledding and sledging with my brother."

Ok. But did you mean sledding? (Both would make sense).

"4) I always put everything I need in a trolley or a plastic basket."

Ok.
"How do you call the shopping card?"

Do you mean credit card or gift card? A credit card is a thing like Visa or Mastercard. A gift card is something you go to the store and buy to give as a gift so people can go to that store and use it there.

Sure! Here are a few corrections to the sentences you provided:

1) I insert a 50-cent coin into the shopping cart and then I do my shopping.
2) I need to buy 200 grams of cooked ham and 150 grams of raw ham.
3) After that, I need to go to the counter for sliced meats and salami, and then to the counter for fruits and vegetables.
In the winter, I go bobsledding and sledding with my brother.
4) I always put everything I need in a shopping cart or a plastic basket.

To answer your question, the term for a shopping cart is "shopping cart." In some regions, it may also be referred to as a "trolley" or a "shopping basket." The specific name may vary depending on the country or region.