One day to explore without Death in Paradise focus

[Vineyard View]

We have one day in October, docking at Point-a-Pitre. We have no experience with Death in Paradise, and no desire to focus on these tours. It seems to be the big focus here from what I can find. 

We are a party of four. Experiencing some of the beauty of the island would be of interest but it appears to be a bit difficult to arrange outside of ship excursions - which we tend to steer clear of because of the larger groups. It also sounds difficult to get a taxi to a place without a tour?

So I am wondering if there are any recommendations of how we might plan our day?  Experiencing some beach time or waterfalls sounds lovely (if not a strenuous hike). Another idea would be going to a small town to explore and enjoy good French food and wines - maybe in a harbor or on a beach?  Any beaches with lounges and shade that serve food and drinks?  Then finally, how to get there and back to port?  

Pictures I have seen of the port are not attractive at all. Reports I Have read seem to support that one needs to get out of the area to experience the island, but the art of arranging that to happen prior to arrival has me baffled. 

Thank you