Just how you establish residency will depend on each state as they are NOT all the same. Typically, residency can be established in 6 months and in some cases is might take a full year. But basically it's where you live and have your mail sent to. Once you establish just where your "residence" is, just how you'll be taxed depends on the state of residence as some will tax income earned out of state even though it's taxed in some other state (it's not common and I don't know which one's they are). For the most part, one it taxed only in the state in which there were earnings and a resident state will credit those amounts. But resident state filing must occur whether there was earning in that state or not.
Here's an example:
California resident drafted out of college sent to Ohio where he signed and took 1/2 of bonus and played there all of Short Season. For that tax year, a California resident tax from was filed for the full amount earned and an Ohio non-resident tax form was filed for the full amount. California credited back whatever the tax was charged inn Ohio. But there was still a small tax liability for California.
The following year, the player moved residence to Arizona at the beginning of the year and played a full season in North Carolina. Player received second half of bonus as a direct transfer which is taxable in Arizona (not California or Ohio this time) and the salary earned while playing in North Carolina is taxed for that state only. . .even though they played games in other states.
Because the team is based in North Carolina, this is what determines that tax base. If the player have moved to a different team that same year, the earnings from that's teams base location would be taxed to that state.