There is no direct impact to your federal income taxes by moving between states.
For state level taxes, you are generally responsible for them based on your state of residency, and states that income is generated from (ex. living in one state and working in another). During the year of a move, this may require filing returns to both (or multiple) states depending on which states are involved.
Ultimately, how this affects you specifically depends on which state you are moving from and which state you are moving to. Types of tax (income tax, excise tax, property tax, sales tax, etc.) and tax rates vary between different states and localities.
For example, some states do not tax income at all (Alaska, Florida, Nevada, South Dakota, Tennessee, Texas, Washington, and Wyoming, New Hampshire only taxes dividends and interest). This article has some good information about this: https://turbotax.intuit.com/tax-tips/fun-facts/9-states-with-no-income-tax/c9RZgthD3##