(Note: I'm speaking from an American-Southwest perspective here. These phrases may be used a bit differently in other places.) "I'm out and about" would be appropriate to describe shopping or doing various other activities out of the house. I don't hear people use this phrase too often, but you're right about the meaning. "On the move" is a bit different. It suggests that someone (or something) is moving quickly, or that there is some sense of urgency associated with their motion. For example, if the police are watching a house because they want to follow the person inside, then when the person leaves, one policeman might say to another, "He's on the move!" There is a sense of urgency because they need to follow him. If you were supposed to go to the airport an hour ago, but you couldn't find your keys, then you might tell someone, "It took me an hour to find my keys, but I'm finally on the move!" There is a sense that you are moving quickly or urgently, in order to make up for the time that you lost. In theory, "on the move" can just mean something like "out and about" (you're away from home, and you're moving from one place to another, not staying in one location). But most people wouldn't use this phrase just to describe someone who is out shopping or running errands. Actually, I don't think I've ever heard anyone use this phrase in real life. I mostly only hear it in films -- where people are much more likely to be doing really urgent things, like chasing criminals.

In that scenario both phrases would work! They have very similar meanings, out and about suggests you're visiting multiple places throughout the day, but on the move suggests you're going between two specific places.