We usually use "make the best of" when a situation is not good, but we can't change it.
eg I'm not happy about this but I will try to make the best of a bad situation so it doesn't spoil my week.
"Take advantage of" is for when you have the chance to get a benefit, usually because of some new thing that happens. So it's much more positive than "make the best of".
eg Somebody offered me a better job and I want to take advantage of it to get a higher salary.
And "make use of" is more common when talking about something that more regular or continuous. It's not for new situations.
eg We have a great library at school, I should make use of it to read books in English.