How would I go about setting up a salary to be paid to be a caregiver?

I suggest you find and hire a payroll company. They will handle all the admin. Stuff. Look at Homework Solutions as one example. Google it...there are a great many

So... you make an agreement (in writing) with the caregiver...set up written job description... written compensation agreement. Next..get a W4. Then..talk with the payroll company... I would guess you tell them how much you pay the person...they send you a weekly statement for the employee..showing all the withholding...and net pay....just like a payroll stub. You write a check to employee...and one to payroll service.