I have always used Metro Car rental, based in Dalaman:
and I believe that regular poster Adana also uses them.
They can arrange a transfer for you and then deliver your car to Göcek later.
Failing that, try this firm:
A Fiat Albea is the Turkish-built version of a Fiat Palio. I hate them!
Seriously, I have used several and always find them under-powered when compared to other cars of their size; I'd far prefer to rent a Renault Clio or a Hyundai Accent.
Children's car seats will be compulsory in Turkey effective 1.6.2010, I know that Metro supply them.
As for a GPS, I am not sure if they are worth the bother. I know that my sister rented one last year, I'll ask her about it.
Edited: 4:14 am, May 06, 2010