We find the easiest way to travel is by car, taking the ferry from Portsmouth to St Malo overnight, from where Noirmoutier is about a 3-3.5 hour drive.
Brittany Ferries run services from Portsmouth to St Malo or Caen
It is possible to get to Noirmoutier by public transport by train from St Malo via Rennes to Nantes, and thence by bus on to the island.
Once you are on the island it is very easy to get around by walking or cycling, so you may find that you don't do much driving except for shopping trips to the main hypermarkets (Super U or Intermarché).
