Each state defines residency on their own. While Colorado says you are a resident when you get a job in CO or live in the state for 90 consecutive days; Ohio says you are when you establish a domicile, or if you spend 7 months (just changed last year from 6 months) of nights in Ohio (called contact periods).
That is when you will be taxed in that state. If you have investment income over a year, you would declare when you changed states and prorate the income. Or, if you have gains/losses on a specific date you would have to figure out (and provide documentation, IMO) which state was 'home' and which got the tax.
So, to answer your question assuming you live off investments only - you would be subject to income tax in each state you lived in based on when the income was earned or a portion of it for non-date-specific income.
Situations like this are very common for snowbirds - folks who head south (many to Florida) in the winter to warmer weather and return to the northern state in spring/summer. Most of these people will claim Florida as their home state of residency, avoiding state income taxes entirely in the northern state.
This is not a black and white situation when challenged - states are keen on this and will expect you to have a FL license, vote in FL, and not be in your northern/income-tax state for more than (usually, varies by state) 183 nights. I've heard of auditors asking for documentation to prove where you were and when you entered/exited the state - which is very difficult to fake (keeping in mind that a tax audit does not have the burden of proof a court has).