I feel a bit confused about using present progressive in the following paragraph:

My family and I are having a good time here in Paris. Every day we are shopping, taking tours around the city. We are eating at the hotel's restaurant where the food is really good.

Please correct my mistakes.
The present progressive here is fine. It stresses the fact that the actions are not routines, but temporary actions happening around the present.
Although they go shopping etc. every day, these actions are only happening while they are in Paris (probably on holiday - lucky them!!). When the holiday finishes, they'll go back to their home and their everyday routines.
After every day you should use the simple present in order to describe the actions you do regularly.
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 Lib's reply was promoted to an answer.
I agree. I can't see anything wrong with what you've written.