Exchange and swap are fine, with exchange being a little more "formal"-sounding. "swap" is a colloquial word, and you might prefer to use "trade" instead...but carries the correct meaning either way. "switch" is another word you could use.
"change" is tricky. It does not mean the same thing as "exchange", but sounds kind of similar. So it is sometimes idiomatic to use "change X with" to mean "trade X with".
Not all cases sound as normal, and the "with" is required to mean this.
I want to say take your partner's book and give yours to him/her:
Here, "change books" alone doesn't really convey the idea that you are taking each other's books. Consider the sentence "Both you and your partner need to change books" - this just says whatever book each of you have, you need to both get a different book...not necessarily each other's.
If you say "change books with your partner", it might be understood because of the "with" (given it probably doesn't mean anything besides exchange). But I wouldn't say that, it doesn't look right to me.
I want to say stand up and go sit down on your friends chair while s/he sitting on yours.
"change places with" is one of the idiomatic cases that sounds normal. However, it does need the "with".
For instance, "Both you and your partner need to change places" has no with, and just suggests both of you need to be in different places than where you currently are.
But "change places with your partner" would be understood to mean taking each other's places. Outside of changing "places with", I don't know what the complete list of "normal" usages of change => exchange would be.
To be on the safe side, don't use change to mean "exchange" unless you're saying "change places with". To be on the even safer side, don't use it then either. :-)