You make the decision of your start day

Take some days off between the job change is a great idea. You can tell your new company (HR or the hiring manager) your planed start date, be sure to get OKed from them. During the whole hiring process, the hiring manager is the person to make decision, the HR is only serve as the middle man.
The other plan can be start work first, take vacation after one or two month. Make sure tell them before you start work. Let them know this is a schedule activeity, the airline tickets has been purchased. They will fully understand.
There will be many up and downs in the life, I am very glad you are back to the work now.
Hopefully, whoever read your story will understand that 珍惜工作 is very important.
Enjoy your vacation and comeback safely!