Whether you choose to self-isolate after a move is a decision that depends on your circumstances. If you socially distance during the process of loading and unloading your belongings, while you travel by car, you may not have much need to self-isolate.
Individuals who are taking trains or planes to their new city may want to consider self isolating for a few days and then getting a COVID-19 test. When you test negatively, you may feel more comfortable about venturing outside and interacting with other people.
It is always best to get information from the CDC website regarding self-isolation and testing protocols.