Might try Frency's in Clearwater. Just down the street from Dunedin on the Gulf of Mexico. Nice littel laid back place, but gets kind of busy on the weekend and evenings. Usually has a live musician or 2 there in the afternoon and weekends. Been there a looooong time.
If you were further south, I'd suggest the Undertow Beach Bar & Grill in St. Pete Beach for "local" lounging and relaxing. It has it's share of tourists, but a lot of the locals in the hospitlatily industry and locals in general, go there to hang. Beautiful sunsets, great drinks, typical beach food (Great Grouper sandwiches!) and usually some live entertainment on the weekends.
The Hurricane Restaurant on St. Pete beach or Gators at the pass in Maderia Beach (Top Deck) have live music, great atmosphere and wonderful sunsets
I'll see what eles I have handy for the Dunedin area.