If I work in Georgia but spend most of my time in California (working remotely). Officially I am located in Georgia. I rent an apartment at both the places - in Georgia and in California. Do I have to pay state taxes on my income tax in both the states?
You are a resident of one state, and a non-resident of the other. Or you could be a "part-year resident" of both. Read the applicable personal tax rules for each state, or hire an accountant to do it for you.
Note that, in the end, each dollar of income is only taxed by one state. It's figuring out which state that gets tedious.