You have to pay state income tax to any state where you work. If you do all the contract work in your home state, you do not have to pay tax to the states where your clients are located. But if you go to the state where a client is located, you have to pay tax to that state on the portion of your income that you earn for work that you do in that state. In most cases you will get a credit on your home state tax return for taxes that you pay to other states. (There are exceptions in a few specific states.)
If you work in multiple states, you should probably find a tax professional who is experienced with multi-state tax filing. The tax pro could also give you some guidance as to how to determine how much of your income is from each state.
