I was already a few times in Turkey, and in such famous and big tourist centres like the region of Antalya/Alanya you can hardly do something wrong if you behave civilized like in every other "normal" country you visit.
There are so many archeological things and ruins to see, you can hardly see everything and it's hard to rate which ones would be the most important ones to see (maybe harbour(s), historical museums, the castle...).
I'm not such a sightseeing person who does nothing but doing tours and trying to see as much as possible. Therefore, most of all, I liked the beaches and the mountains in the landscape all around me
But if you have the chance to go Istanbul and if you never have been there before I would recommand to visit it, but the problem is that it is awfully far away from Alanya. It was a must see for me and I didn't regret it, but not the easiest thing if you are basically located so far away.