Things I have done in recent days (A native English speaker would not say "these days" to refer to the past, even the very recent past.)

It has been many days since my last notebook entry. (If you were writing about a real book, "notebook" would refer to the whole book. A notebook entry would refer to one section.) In recent days, (You should use a comma when you put the time phrase at the start of the sentence.) I have been with my family, my friends and other relatives from Argentina. They were in Santiago and Valparaiso. there They spent the New Year there with fireworks by/over the sea. 4 days later, they traveled to my city and we had a family dinner. We were (unless you were eating for 2 days! :)   ) with them for 2 days then they returned to Santiago to go their home in Buenos Aires.

(You don't want an article here) Last week I was at Santiago visiting the doctor. I went with my mother and my 2 sisters in my aunt's car. After visiting the doctor, we went to eat in the to Costanera Center. After that, we saw a lot of smoke because we have many forest fires in the country. The air was heavy. I resented (I'm not sure what you mean here; perhaps, I "rested" my eyes.) my eyes, but we went to take a bus to return. Finally, I had good results in my medical exams.

Gracias por su ayudar con mi escrito. Felicidades en su buena noticia.