First off, don't confuse it with the small resort town of Ovacik near where the Lycian Way begins. That Ovacik is a few more hours west and the mountain you'll want to be climbing there is Baba Dag (an easy mistake to make when you're planning your trip via Google Maps). Anyway, this Ovacik is a 2,004 m (6575 f) peak most readily accessed from the village of Ovacik to the east, although the northwest slope may be the easiest grade to climb. Either way, you'll need to put aside more than half a day to ascend, partially because you may need to double-back at points due to a lack of plainly designated trails. Don't worry, a little patience, endurance, and ingenuity will reward you with a summit view.
If you find life much more satisfying going downhill on two wheels, then you may as well go with the other popular option on Ovacik: mountain biking. Lukka outdoor conducts a guided descent from along the plateaued shoulders of the peak through the Kesme Valley, with an option to extend the ride another 15 km to Alacasu Bay on the Mediterranean (just south of Çamyuva).
If you'd just like the grandeur of an incredible view, skip Ovacik and go to Tahtali. If you're looking for an adventure experience while in Antalya, then Ovacik should be on your list.