Things to do, if I can remember as it was 2001 we were there.
Down on the habour they run "donkey island trips" there was a big double decker boat which seemed to have more of a party atmosphere than ours but the trip was still better than the similar one from Kusidasi. The Google Earth resolution is quite good around Cesme.
There is a heated spring at a hotel, it has been harnessed into small swimming pools, but a nice little trip.
With a car you can drive down to Kusi fairly easily either round the coast or back via Izmir. To the North of Kusi you have Ephesus for culture and AquaFantasy and Adaland for fun.
On the boat trip we saw nice beaches to the north of Cesme, but we were at the Club Pomelan just past Ciftlikov. Keep going past the hotel and the road forks, there are nice beaches at both ends of the road.
There is a big street market in alicate (check spelling) and the old bazaar area in Izmir is worth a view. Don't buy the perfume! There was a really nice chap flogging good perfume from his shop in Cesme and also he said hello in various markets. Unfortunately the % was more genuine eau de Toilette than eau de parfum. ( excuse the French)