To use the public hospitals for regular visits, I believe you need to "join" a particular regional clinic, and to do so you need to make an appointment that could be at least a month or few in the future. I think they are free, at least for locals. The architecture of the public hospitals is often open, the rooms don't have glass windows, but these open kind of Trini blinds... not enclosed, air-conditioned sterile environments.
I personally have been going to private doctors and specialists, who work from private hospitals or clusters of doctor's offices. They are ok, for example, St. Clair Medical Centre, Doctor's Inn in POS, Medical Associates in St. Joseph, and St. Augustine Private Hospital.
I have looked into health insurance with Sagicor only, and it seems that you always have to pay monthly, there's one plan that's about TT$10,000 for a year... so that's a commitment. And they seem to only partially reimburse your care.