How do I set up a pay system for a private caregiver?

I hope you do not expect full day or full time care for $1500 a month.
$1500 a month will get you a few hours a day for about half of the month.
In many areas minimum hourly rate for a caregiver is $20 to 25 per hour.
And Room and Board is not considered compensation for work.
Just had to get that out there for you.
The software route might be the least expensive way to go.
Companies that will do all this for you do charge a fee for the service.
If you have your taxes done by a CPA they might be able to help you with this as well.

Here you go...It might be easiest to download an app or software for this, or see if your bank offers anything like that to assist. The IRS website might have some information as well.
https://due.com/payroll/setting-up-payroll-for-your-business/

I hope you're not expecting many hours of caregiving for such a small amount of money.

Consult an accountant or CPA to set. A small local payroll company could also help you.