You would be a part year resident for each state. In the beginning of the interview process under Personal Info, you are asked if you made any money from another state. By putting your new address, the program will automatically pick up the state you have moved into.
The states, once you select part year resident, will ask the dates you moved into/left the state. At the bottom of each W-2 you should have the earnings allocated to that particular state. If you moved more than 50 miles and you meet other qualifications for moving expenses, the state you moved into sometimes allow you to deduct moving expenses carried over from the Federal.