Summarize:

After you make your first hit, you will need to reduce your target area to the spaces surrounding the space that was a hit.  Since ships in Battleship range from 2-5 spaces long, it may take you several turns to sink the ship you have hit. Begin by striking above, below, or to one side of the space you have hit to discover and hit more of the ship. If one of your strikes is a miss, try the area on the opposite side of the space that was a hit. Keep striking until you have taken your opponent's battleship. You will know when you have taken your opponent's ship because players are required to announce when a ship has been sunk. After you have sunk the first of your opponent's ships, you will need to resume firing randomly (or in the center of the board) to find another ship. Then repeat the process of firing around the hit space until you have sunk another ship. Playing the game this way will reduce the amount of turns that it takes to sink all of your opponent's battleships and it will also increase your chances of winning the game.
Reduce target area after you make a hit. Fire around the area of your hit. Repeat method to hit more of your opponent's ships.