Take a taxi if you have a complicated journey, or you have heavy baggage, or it is raining. At night it can be a lot cheaper than getting robbed. Check exact route before starting, eg east taxis are either San Juan or all the way up the Eastern Main Rd to Arima, or highway , which is faster and runs closer to the airport but misses places like Tunapuna and Curepe. Taxi despatchers find taxis for new arrivals, ask to see the rate card for taxi fares to different places. Unlicensed taxis outside the main parking area charge less, depending on the volume of business.
Public transport is much cheaper.
Coming back to the airport do the same in reverse. People are very helpful if you need to ask. Tickets not available at the airport. From the central bus terminal at the old railway station you will have to walk to Independence or Woodford Square for a route taxi for your ultimate destination. This is not advisable at night, especially if carrying luggage.