Is there an airport that you have to fly Into? Where will you be before Sicily..Rome? Does the six days include the day or arrival?
Late September/early October is too late for lounging/sunning on the beach.
You might get a great day, but no beach clubs will be open so you'd have to bring your own towels, chairs, and probably food. Of course you could so some lovely beach walking, but I would not make the beach one of your two bases with only six days on the island.
You'll find great pastries everywhere, and of course, locals life everywhere.
I think you need to do some further reading in order to pick the bases that would appeal most to you. You probably ought to choose either the west side of the island, or the eastern side...the latter is more popular with first-time visitors.
Do you want one base to be in a big city--Palermo or Catania?
Or do you prefer a smaller city or even a village?
Or a farm stay in the countryside (need car for that)?
The major ruins are outside Agrigento but there are also important sites in Segesta, Selinunte, and others. And Taormina has a Greek theatre dating from the third century BC. Maybe that's enough to satisfy the "ruins" requirement.
French is not very useful but I would not worry at all about getting on without speaking Italian. Everyone is the hospitality business is likely to speak English--restaurants, hotels and B&Bs. And even in a rural farm (agriturismo) there will be someone around who will be able to communicate in English.
I suggest getting a good guidebook--maybe the Rough Guides--and narrow down a few places that appeal to you..then come back with questions...
