A gym without dumbbells or weights?? That's weird, which gyms do
you go to? Anyway, the reason why your wrists hurt is probably
because you either set the machine too heavy or you are using a
wrong grip that cause you to bend your arm whenever you are pushing
the weight. I agree with eagle, just find a better gym with
weights.If not, either use the chest machine with a lighter weight
to get better form or just stick to good old push-ups, dips and
pull-ups (they all train chest).
Btw,chest expanders are quite useless, unless you are already
quite advanced in bodybuilding and just wants to tone up....