The general rule of thumb for state taxes is: you pay state tax on state source income (from working or rental property) in the state the income originated, if you do not live in that state you file as a nonresident. You pay state income tax on all income, regardless of source to your resident state. To avoid double taxation on income at the state level, your resident state will give a credit for any tax you paid on income in another state.
For a part year resident, you allocate the income exactly the same way, just using the dates to determine your resident/nonresident status. You are correct, you will file as a Part Year Resident for both CO and WY (technically for CO you will initially determine Colorado taxable income as though they are full-year residents then subtract out the retirement/social security you received after you moved to WY.) WY will include retirement/social security you received after you moved to WY and the rental/royalties, but you will receive a credit for the tax you paid to CO for the time you were a nonresident.
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