Fighters eat bombers because bombers arent agile and have low fire power, therefore acting as an escort for un-agile ships.
Bombers eat cruisers, since they are made to bomb bigger, slow ships such as Cruisers are. However they cant defeat a battleship (unless highly consentrated) due to its superior armor. They quickly fall to fighters.
Cruisers eat Battleships (In single players Cruiser destroyed a Battleship/Destroyer)
Battleships eat all the small ones and other battleships. (Biggest firepower used to destroy enemy fortifications and bases)
Not sure where Gunboats stand here. Well they are supposed to counter fighters, but may also inflict damager on bigger ships. Hard to say how they interact with Bombers... No experience here.
P.s. whats a diffrence between a Battleship and a Destroyer?