The ferry to Port Aux Basques on the south west coast runs all year. Like others have said you can boondock until the parks open in May, usually on May long weekend.
It's recommended that you book the parks, since they fill up quickly. Also book the ferry trip across. The trip to the east side of the island (Argentia) only runs May to Sept and is a 14hr trip. The run to Port Aux Basques is about 8hrs. During the summer tourist season the boats can also be busy.
What do you like to see when travelling? There is lots to see here from history to festivals. There is also fishing, scenic site seeing whale watching.