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Antalya is the largest nearby city which has a pretty old quarter and is worth a visit.
There are frequent buses between Kemer and Antalya which cost €3 each way. Get an "Express" which takes about 1 hour. There are also buses which go via Goynuk but these take 1½ hours.
The bus station in Antalya is a long way from the city centre, so most people get off the bus at an unofficial stop which is characterised by a scruffy blue walkway overpass across the road. From here walk round the corner and get any dolmus going to Kale Kapesi, this cost 1.15YTL in summer 2007. As you get close to the city centre you will see the striking old minaret tower, just get off when everyone else does.
Head to the old quarter called Kaleci for some peace away from the bustle of the busy shopping street. This is enclosed on one side by Hadrians Gate and on the other by the clock tower and the old historic marina. Wander through the winding streets and restored ancient buidlings in this charming area and maybe stop for a bite to eat in the many restaurants or pop into one of the museums.