Rent a sand buggy and ride the south coast dunes of Natal's shores from fishing village to fishing village. Pirangi do Sul, Tabatinga, Tibau do Sul, Pipa, and Baia Formosa are separated by miles and miles and miles of empty and idyllic beaches.
These little spots of human presence in the wilderness of northeast Brazil hardly ever see any foreigners, so do not be surprised if you are the only one. But worry not, the locals are very relaxed and friendly enough to help pull the buggy out of the sand if you happen to get stuck. This trip is best taken with the chill approach to life that is characteristic of this part of the globe.
I found it particularly interesting that, having stopped in the middle of nowhere for a dip in the sea, a man with a styrofoam cooler box appeared out of nowhere, bearing ice-cold coconuts. For 3BRL (less than $1.50) each, we ended up making his day and got his whole stock.
My recommendation is to set off in the early morning hours, and catch the sunrise at five o'clock in the morning already outside of the city. In the small villages that I mention above there are plenty of little eateries, and down close to Baia Formosa there are a few beach restaurants serving fresh sea catch.
Do not forget your towel as you will need it for the beach and to cover your head when the 36º tropical showers come around.