Texas does not have a personal income tax so there is no state return required for Texas.
You were a part year New York resident so you report all of your income and expenses on the NY return while you were a resident.
If you are going to be using the TurboTax desktop editions to be installed on a personal computer then you should purchase and download the Deluxe desktop edition. You receive one free state program download with the Deluxe edition - https://turbotax.intuit.com/personal-taxes/cd-download/
Into the weeds a little - When using the TurboTax program you enter your personal information in the section called Personal Info. You will be asked for your resident state and you should select Texas from the dropdown. Further on you will be asked if you lived in any other state and you should indicate Yes and choose New York and enter the date you moved to Texas. You will also be asked if you made money in any other state and again you need to choose New York.
