Yes, you have to enter state tax withheld. State tax withheld from your pay should be in box 17 of your W-2. You enter it on the same screen where you enter your other W-2 information, in the federal section of TurboTax. The state information is in boxes 15, 16, and 17. You might have to scroll down a bit to see those boxes. The state information will flow to your state tax return.
You have to get the information from the W-2 form from your employer. You can't figure it out yourself or use your pay stub.