Firsly,there are no good beaches near Port of Spain..try Maracas Beach on the northern side of the island, it's very popular. You should eat : doubles, roti, pelau, callalloo, shark and bake at maracas....all very tasty! u can also try other creolae and indian dishes...i just listed the major ones. there are also shops near maracas or at maracas beach that sell local candy and preservatives...those are really god too. u should see Tobago(there is a daily ferry),the botanical gardens, mount st benedict is also nice,the pitch lake(if u can get there), san fernando hill and generally tour Port-of Spain (shops on Frederick Street, museum, savannah etc).
do go to a local play..u might find it interesting...unfortunately Carnival is over which is the main time for steelbands when they practise but u may be able to hear one depending. do enjoy ur stay:) (that's very important).
To answer another of the answerers,Trinidad and Tobago is not crime ridden, it is just necessary to take the same precautions that any reasonable person would take in the US, UK, Europe etc. just because we are in the Caribbean does not mean u can leave ur doors open, walk around flashing jewellery or money, walk by urself at 12 in the night etc.