Always control your details. I agree that Uber or other public transportation is your best bet on a first visit. Get to know the woman a little bit before going all out on a rental car. If you all become friends, then yeah, jumping into the rental car thing makes sense. You can toss her the keys and kick back and relax as she steers you around the island. But right now, you don't even know her, or know if you'll even like her. Also, Santo Domingo can be tough to drive in. I mean, imagine dropping into Manhattan and you can't read or write the language.
Again, in my view for your upcoming trip: Uber, driver or taxi. One step at a time, young fella.