Well, if you've never been to Istanbul before, it's well worthwhile to spend a few days there. Plenty of places to see and things to do, depending on your tastes. Perfectly safe for nice single ladies, and probably you too
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Dead easy to get from either airport into the city centre using the Havabuses. And cheap too. For instance Sabia Gokcen (SAW) to Taksim is just 18TL. They run roughly every half hour from 4am until 1am.
Have a look on one of the hotel booking websites for hotel prices in your range. Many of them also do free or very cheap transfers.