Move away: To move away means that you leave one place to go and live in another.
'I was born in London, but we moved away to Liverpool when I was very young.'
Move in: To start living in a new house. It is often used with to when we want to show which place.
'I bought a new house. I'm hoping to move in sometime next month.'
'I'm going to move in to my brother's apartment.'
Move on: To change from one thing to the next thing. To stop one thing and start doing another.
'I think we have discussed this topic enough. Let's move on.'
'I've been working here for ten years. I want to change my job. It's time for me to move on.'
Move out: To move out means that you leave your home forever. You move to a new home.
'I had to move out when the rent became too high for me to live there.'
Move over: To move to the side.
'I want to sit on the sofa too. Can you move over?'