In home daycare. Do I enter vehicle expenses & miles driven under business, personal, or both categories? Is this something that I need to claim the same way each year?

No, you should only enter your home day care expenses in the Self-employed business section of the software. Only when you are paid with a W-2 as an employee, should you enter unreimbursed employee job expenses. 

You enter all of your expenses for self-employed work, with that income (you can have more than 1 self-employed business). To enter your driving expenses:

1. Head into your return and select Federal from the side bar 
2. Then select Income & Expenses from the categories along the top of the software 
3. Scroll down and select Edit next to Self-Employment 
4. Select Edit next to your daycare self-employed business (or you'll begin adding it if you haven't already)
5. Click Continue through the informational summary screens
6. You'll come to the Income and Expenses summary screen, here you'll see a listing of all the types of expenses you can enter including miscellaneous. Anything spent for the business qualifies. 
7. On the summary screen, scroll down under Expenses and click Start next to Vehicle to enter your driving costs