Battleship is the game everyone played as a kid and few have played as an adult, which is a shame because there's a genuine probability puzzle hiding underneath the kids'-game packaging.
The single biggest discovery for intermediate players is parity. Battleship ships are all 2+ squares long, which means at least one of every two squares on the board must be empty. If you only target squares matching one parity (think of a checkerboard pattern and only shoot the dark squares), you're guaranteed to hit every ship in half the shots.
Ship placement is the other half of the game and is consistently underrated. Most casual players cluster ships in corners thinking opponents won't shoot there — which is exactly where opponents will shoot once they realise parity sweeps cover the board evenly. The veteran placement is to scatter ships across the board with at least one square between any two ships.