I am a travel nurse. My state of residency is Georgia. Last year I worked in North Carolina, Georgia, and Colorado. How do I file my state taxes?

You will need to file a full year resident Georgia return as well as nonresident NC and Colorado returns. 

You should prepare your nonresident Colorado and North Carolina returns first and then your Georgia return.  That way the software can calculate the credit on your Georgia return for the tax you pay in your nonresident states. 

You will get a credit on your Georgia return for the taxes you pay in other states, however you do not get a dollar for dollar credit.

Excellent article  on this subject:

https://www.americantraveler.com/state-tax-questions