How do I handle taxes if I intend to live part of the year in two different states (e.g. Owning a home in a primary state, and renting an apartment in a secondary state)?

I am an established primary resident in one state, but the nature of my work takes me to another frequently. Because of this I would like to rent an apartment to reduce the expense of hoteling while in the second state. Effectively I would be living in two different states as my needs require, similar to a 'snowbird' who moves to warmer climates in the winter. My company would leave my established residency in the primary state.