Do I need a wire transfer routing number or a direct deposit routing number to get a direct deposit to my bank account?

No, definitely Not a wire transfer number.

To direct deposit a tax refund you must use a US bank routing and account number for a bank in the US.  The bank information for direct deposit of tax refund is entered in the File section of the TurboTax program on Step 2.

use the routing number printed on the bottom of your checks.