Hi, I'm Susan, and I work for TurboTax.
Some states provide a free state e-file website - check your state's official website, usually the two letter abbreviation with the .gov domain name. For example, I live in California so it's ca.gov.
You likely will have to file multiple returns, and exactly which returns will depend on which states are involved and what their rules are. Generally, when you live in a state for part of the year and have income earned while you lived there, you file a part-year resident return for that income. When you live in one state and have income from a different state, the situation gets more complicated. You may need to file a non-resident return for the state where you earned the income... OR you may need to include that income on your new resident state return - some states have agreements to ensure that income is only taxed once. Without knowing which states are involved it's hard to be more precise.
Hope this helps a little, at least pointing you at the research you'll need to do next.
Susan, TurboTax Tax Compliance Service manager