Corrections in red.  Stylistics and variants in blue.

My Day

Today I (will) sign the contract, and on Wednesday I (will) start to work. [1]  I'm a little nervous, because I haven't had a job for a looong time. But I'm also happy to start this new experience, and for me it will be a little holiday from my children :-)) Next week I will register my children at camp, so they can play with other children, and when they come back home, they will go immediately to bed ))) In Italy the weather is unstable.  In the morning, we must put on a sweater, but in the afternoon, the weather is warm. It is very bad weather, because we have always influence.[?? 2] So I hope that the summer begins soon, so we can go to the swimming pool. What is the weather like in your country?

[1]  The word "will" is optional.

[2]  Do you mean, "we always have influenza"??

Good luck at your new job.

Where I live, it's hot and getting hotter by the day.   "by the day" =  gradually and steadily