If you need to file in more than one state, you will need a state download of software for each state. If you are using TurboTax Online, each state download you need costs additional if you are using Deluxe or higher. One free state is included in the Free Edition.
Otherwise, you can file states with any version of TurboTax. Whether you need to upgrade depends on the types of income and deductions you have.
You can use this comparison tool to evaluate different versions.
See also this help article for TurboTax Online and this one for TurboTax Desktop.