Do I pay city local taxes the year I get of the military?

Generally, the city/local taxes are based on where you physically lived and/or worked.  So once you left the military and found a new employer then applicable city/local taxes would be applicable depending on the location of your residence or your employer location.  

As you complete the state interviews you will be asked residency questions; those answers will generate any city/local tax forms that TurboTax handles for you.

Additionally, when you entered your form W2 information if there were local taxes withheld (boxes 18-20), you get credit for those on any state/local forms as well.