People who are on probation are typically required to check-in regularly with their probation officers, sometimes by phone, sometimes in person. In addition, they may be required to attend counseling or classes that address substance abuse; and they may be prohibited from traveling.
If you want to move while on probation, the issue will be to what extent, if at all, the move will affect the responsibilities you’ve undertaken when accepting probation. For example, moving from one unit to another in the same apartment complex will probably not affect your reporting duties; but moving to another city, and certainly to another state, will have serious ramifications. The local court and probation office will not expect to supervise you from afar; and they’re not going to let you off the hook just because you’re no longer local.
Moving beyond the jurisdiction of the court that sentenced you (and the reach of the probation office that supervises you) will normally require transferring supervision to another county and office. Your local probation office is not going to do that lightly; it’s time-consuming and expensive, and you will need a very good reason for the move in order to convince the office to undertake the transfer.