I'm not a teacher, but "listen for a change" is telling someone to pay attention to the speaker, and it further says that person being told to pay attention does not normally listen. So, if they do listen this time, it will be "a change", a difference in behaviour from their usual pattern. "Get on my good side" means, "do something to earn my favour", "act in a way that pleases me".
Stuart's almost certainly right, but without context it's hard to tell. There's another possible interpretation for 'listen for a change' - as here: 'Government rhetoric for the last five years has been all about cracking down on crime, but there's a new Minister at the Department now, and he's got lots of new ideas; so listen for a change' - that is, expect a change and pay close attention to (listen [out] for) any sign of it.
In the more likely meaning that Stuart described, there'd often be some form of punctuation. 'You always do something different from what I keep telling you. Why don't you listen - for a change?'