I live in PA, work in DE. Which state is receiving the payment in Box 17 of my W2? If it's the DE income tax, do I have to file separately for PA?

The tax withheld is reported to the state which is listed in box 15.  Which state is reported in box 15 of the W-2?

Delaware defines a nonresident as follows:

Nonresident

Nonresidents are individuals who are not Delaware residents at any time during the year.

Pennsylvania defines a resident as follows:

Resident 

Defined as a person that is either domiciled in Pennsylvania or has a permanent abode in Pennsylvania and spends a total of 183 or more days in Pennsylvania.

In TurboTax Online, you likely will file a full-year Pennsylvania resident tax return and a nonresident Delaware tax return.

Delaware and Pennsylvania do not have a reciprocal agreement.

If you live in PA and work in DE, then Yes, you definitely have to file in both states.  You must file a non-resident DE tax return and your home state PA return.

You'll be able to claim an "other state credit" on your PA return for the taxes paid to DE, so the net effect is that you won't be double-taxed.  But you still must file a return in both states.

In TurboTax, be sure to complete the non-resident state return first, before you do the home state tax return, so that the program can calculate and apply the credit.