Where should I go spearfishing?

The first essential is water clear enough to see the fish. In practical terms this means 10' or more (fish can be speared in much, much less, but there is no point making things hard for yourself when you are starting out!).
Try and find an area where there is not too much boat traffic, and find out what the tidal flow is like. Most bays are fairly sheltered, but headlands will often experience significant currents .
Rocky areas with plenty of weed will normally hold the most fish. Stringweed beds often hold bass and mullet. Talk to local fishermen about where fish are to be found (but always take their answers with a pinch of salt)Some areas will only hold fish on particular states of the tide. As a very general rule, fish moves into bays and estuaries on the rising tide, then out to headlands and offshore reefs as the tide falls
For a first attempt at spearfishing a sheltered bay, with plenty of rocks, on a rising tide is probably best.