Do I pay taxes on money paid to me by my grandfather for caring for him? If so, how do I file?

Yes, you would pay taxes on the amount that you receive to care for your grandfather. You won't be taxed as an employee, but will record the income on Schedule 1 of the 1040. 

To enter in TurboTax:

1. Go to Income & Expenses
2. Click on Less Common Income
3. Go to Miscellaneous Income
4. Click on Other Reportable Income
5. Answer Yes to Do you have any other taxable income?
6. Enter description and amount (Caretaker for grandparent)

This amount will be entered on the 1040 as Other income. You will not pay self-employment taxes on this amount. Ideally, this income should be reported to you next year on a 1099-NEC.