You may need to see a professional. I believe that in general, Turbotax can only prepare 3 (or maybe 5) state tax returns for one federal return. If you have income in several states in the same year, you may need professional assistance. (If you mean you just moved a lot, and have income from 1 or 2 states per year but they are different states, that's easier.)
If you didn't file for 3 years and expect a refund, it should be paid as long as you file within 3 years of the original deadline, and you generally won't be assessed late fees. But if you owe taxes in any state, you will probably owe penalties and interest. If the amounts are large, a professional may be able to help you negotiate a lower penalty.
In general, you must file a state tax return for the state where your permanent residence is located that reports and pays tax on all your world-wide income. If you earn money in other states as a non-resident, you usually file a non-resident for those states to pay tax on income earned in those states, and your home state will give you a full or partial credit toward the tax you pay in other states. As I said, I believe Turbotax can only prepare a limited number of non-resident returns to go along with your home state return, so you may want professional assistance.
*Answers are correct to the best of my ability at the time of posting but do not constitute legal or tax advice.*
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