The following is a comparison of different things which will tell you the overall pros and cons of each, while comparing them to other types of ships.
The Battleship is the most armored ship type in the game. It, unlike the Destroyer and Cruiser, can take multiple hits befor being sunk. However, while this can help you, it will also inhibit you. Speed relies on how heavy something is when it the water. Battleships rely on heavy armor, guns, adn equipment. Although strong and reliable, they will be slow AS HECK. They will be most vulnerable to torpedos. SO, best be stayin' out of the way of those things!
Battleships are also great for distractions from other battleships. Cruisers will be the best antidote for a situation involving battleships. These things are basically Heavy Aircraft (Bombers). They are basically slow moving behemoths that need to be alive as long as possible, as they are crucial to the mission.
Aircraft Carriers, historically, are basically mobile bases. These things are able to take hits from anything, yet, like all other ship types, are literally paper when it comes to torpedoes. These things, unlike the Destroyer, can take at least 3-8 hits from regular cruiser guns. Destroyers and Carries match up well in a team, yet they will apart when facing off against each other. Aircraft Carriers will be a good wall to keep the Destroyers in behind, but Battleships will work A LOT better.