In
2010 from January-October I lived in Arizona and received unemployment
From those months. In October of the same year I moved to Michigan and
Got a job, so I do have a W2 from working in Michigan and that is the
Only W2 form that I have. So would I have to file for state taxes
In/for 2 states? Bear in mind that my unemployment that I received had
No federal or state taxes withheld for the period.

Congrats on getting a job! You will have to file part-year resident returns with both AZ and MI. You pay AZ income tax on your income received while living there (Jan - Oct), and MI income tax on income you received after moving there (Oct - Dec).