Turkish friends of ours spend every Sunday lazying around at Ada, Sovalye Island. They love it there and always have pictures on Facebook.
Lots of good places to try at Calis, we all seem to find our favourites. Vojo's, Georges (our favourite) Casa De Mama, Bambu (good for cocktails as well). Eyna for real coffee any time of day and of course, the 3C's Charity Shop in the basement. If you want to go further afield, the Istanbul in Kayakoy will pick you up on Saturday night for free and they have a really good meal, entertainment and dancing. In Fethiye, you have mentioned the Mozaik Bahce, Cafe Cafe is great for real coffee and a cake, not far from D.&.P Perfumes where you must go and try the excellent copies of all the famous perfumes at really cheap prices.
During the day do not miss Cafe Park Teras. This is at the new Marina Park in Fethiye just where the car park is. There is a chemist on the corner, if you go next to the chemist there is a building with a lift in it, get in and go to the top floor, it's at the top of the building on the roof and has panoramic views of Fethiye and Calis. Get a beer or a fresh orange juice (my favourite) and relax. We love this place. This is a write up from the excellent Turkey's For Life
http://www.turkeysforlife.com/2012/08/fethiye-terrace-bars-cafe-park-teras.htmlAlso, along the Marina passed the Cafe Park Terrace there are lots of restaurants, We love the Shaka have lunch or go for dinner at these restaurants, they often have entertainment in the evenings.
Sundial is lovely, but they will not pick up unless there are four or more people. It's quite hard to walk to, so if you don't have a car a taxi would be best.
http://www.sundialotel.com