Hi sickofnosun (aren't we all!)
As breezeclose says Pefkos is quite flat in the resort centre and does have a nice beach but there's not alot of sightseeing to be done. You could visit Lindos (5-10 mins by taxi) and if you get dropped off at the main square the centre of town is pretty flat but if you want plenty to see you'd need to hire a car and get out and about.
Faliraki has a lovely stretch of beach but again not alot of sightseeing in the vicinity and early/late season it can be a bit of a ghost town.
I'd go for Rhodes Town, being on the northern tip the beaches are pebbly and windy but there is so much to see and the majority is pretty flat. Rhodes Town, being the capital, also has a certain buzz whether it's peak or low season. As i said the beaches aren't fantastic but you could pick somewhere with a nice pool and maybe get a bus/taxi/hire car to one of the better beaches on the island.
October is a little more unpredictable than high summer but generally it's warm enough to sunbathe during the day though the evenings can be quite cool.