Generally, when you live in 3 states, you would file as part-year residents of all 3 states. However, there are some exceptions in some states, such as if you maintain a home there all year and spend more than 183 days in the state, you would file as a resident which means all of your income earned in all states would be taxable.
In the beginning of TurboTax, in the personal information section be sure to answer that you lived in more than one state. It will only allow you to enter one extra state in that area. When you get to the state section you will then be able to select the 3rd state if it is not already listed. As you walk through and answer the questions, TurboTax will help you figure out what you need to do for each state.
If you have state wages for the state, it will generally show in the list of states.