to leave the place where you are staying and go somewhere else

I've lived in this town long enough - it's time to move on. 我在这个地方已呆了很长时间——该搬家了。

to start a new activity

I'd done the same job for years and felt it was time to move on. 我多年来一直在做同一份工作，觉得是该换换工作了。

to accept that a situation has changed and be ready to deal with new experiences

Since he and his girlfriend broke up, he's been finding it difficult to move on. 自从他与女朋友分手后，他发现自己很难接受现实开始新的生活。