We are considering a gulet trip that ends in Demre on or around 9/24. One can apparently not rent a car in Demre (at least according to Kayak), so we're looking for transportation to Kaş. From Kaş we would like to visit some of the sites in the area. Then, after 3 nights in Kaş, we need to get to Antalya, from which we'd also like to visit local sites. What are our options short of a private taxi? I'm not finding much online for a local bus or dolmus. Before committing to this gulet, we'd need to confirm schedules and availability of transportation. TIA

Don't worry! There are tons of buses and minibuses along these itineraries.
I suppose you will leave your gullet an Andriake harbour.From there you can go by (rather cheap) taxi directly to Kas. Otherwise take a taxi up to Demre Otogari and board one of the frequent minibuses to Kas.
Direct buses between Kas and Antalya run every 1-2 hrs. Buy the ticket 1 day ahead at Kas Otogar.
Minibuses go up to Kumluca (when they are full). There you can find frequent minibuses to Kemer - Antalya.

Antalya - Perge: tramway
Antalya - Side: bus or mninibus
Antalya - Phaselis: Kemer - Tekirova bound bus (every 20 min or so)
Antalya - Aspendos: bus plus taxi or all by taxi
Antalya - Alanya: frequent buses

This is really helpful, neckervd . We so often use rental cars and can just go where we want when we want, so navigating a foreign transit system without a real schedule or route map is a bit daunting. Also, we're still researching all the places in the area to try to see, so figuring out trasport to them is an added wrinkle to iron out.