If its where I think you could walk into Kremasti from there, which is a lovely village (or Trianda if you are feeling very energetic) and a there is a regular bus service to get you round the island. If you take a short cab ride into Rhodes Town you can get lots of day trips on boats in the harbour going to Lindos, Symi etc.

Electra Palace is closer to Trianda than Kremasti, there's plenty of bars and restaurants within a 5 minute stroll. The only place of major interest close by is Filerimos, through Trianda and then quite a way uphill (so you'll need a taxi) but there's a monastery, a few ancient ruins, small church and a memorial/cross that you can climb the stairs for the most fantastic views.

Rhodes Town, especially the Old Town is fascinating and a short bus or taxi ride away. Buses don't pass by your hotel but it's a short walk to pick one up on the main road. As Ninkynonk says several boat trips availble from Rhodes Town and buses for the East coast.

Rhodes in October can be lovely, sea has warmed up by then.

Definitely head to Rhodes Town old town as your so close, also consider the catamaran to Marmaris.
Lindos nice but a long trip.

Thanks for responses.

My next question was can you go a trip to Turkey, so that answers that for me!! That's wonderful.