Using the civlopedia, here's a unit guide for your use:
ANTI-SHIP MISSILE
Effective as a one-time use weapon, cheap, and effective against any ship; can be carried by Cruisers
Counter-Unit: Can be vulnerable to Destroyer Intercept
Best-Use: 75% chance of evading interception; when facing an overwhelimg stack approaching your city, CHEAP, FAST, EFFECTIVE in reducing the enemy stack's power while moving/repositioning city defense. Also, VERY effecting against "spotted" stealth ships.
BATTLESHIPS
Effective as most powerful ship in the game
Counter-Units: Vulnerable to Missiles and Bombers
Best Use: ship-to-ship Attack; bombarding planetary defense
BOMBERS
Effective against most ship types
Counter-Units: Vulnerable to Destroyers and Fighter Squadrons
Best Use: Recon, Bomb ships, Bomb City Defenses, bomb tile-improvements
CARRIER
Effective with Fighter and Bomber Squadrons; Direct Combat capabilities are highly limited
Counter-Units: Battleships, Missiles, more powerful ships
Best Use: Carries multiple aircraft squadrons for defense against attacking bombers/fighters and part of a city attack group
CRUISER
Effective as medium-strength warship; can carry Anti-Ship Missiles and Doomsday Weapon (nukes)
Counter-Units: Bombers, Missiles, more powerful ships
Best-Use: Can bombard city defense as well as carry Missiles as part of a city attack group
DESTROYER
Effective only as a mediocre front of the line ship, powerful against missiles, Fighter/Bomber Squadrons
Counter-Unit: Missiles and more powerful ships
Best-Use: Can Intercept Missiles and Bomber/Fighter squadrons (best with air-intercept promotions)
FIGHTERS
Effective as protection of Star Systems and Bomber/Fighter Squadrons
Counter-Unit: Air-Intecept promoted Destroyers and Fighter Squadrons
Best-Use: Recon, intecept Figher/Bomber squadrons; destroy tile improvements and bomb city defenses
INVASION SHIP
Effective and designed to capture enemy Star Systems (slow and weak in other battles - no defensive bonuses)
Counter-Unit: PDS, Bomber/Fighter Squadrons, Missiles and more powerful ships
Best-Use: First-Strike; +200% city attack
PLANETARY DEFENSE SHIP (PDS)
Effective and specialized to defend Star Systems from attack
Counter-Unit: Invasion Ships, Bomber/Figher Squadron, Missiles and more powerful ships
Best-Use: +200% planetary defense
STEALTH SHIP
Effective for invisible scouting and destroying improvements
Counter-Unit: NOT INVISIBLE to enemy Stealth Ships or scouts; when "spotted" by an enemy Stealth Ship or Scout, very vulnerable against Fighers/Bombers, Missiles, and other more powerful ships
Best-Use: Sentry/Scout against approaching attacker, destroying tile-improvements and construction ships, and hidden nationality. Can also be used to guard/scout against enemy stealths approaching or in your own territory.
In FF, as in the real world (and most of civ) Each ship (unit) has an equal counter. It's ABSOLUTELY best to use a combination of ships in your attack stack (as well as to defend your city)
As Earl pointed out - PDS's seem very strong to him (in fact a very few PDS's in a city can DEFEND and win against a battleship). Put a an equal number of invasion ships BENEATH the battleship (as well as a destoyer or two to protect it from enemy aircraft) and the PDS's will lose.
note: invasion ships are rather useless in attack anything other than units IN a city, just as PDS's are useless OUTSIDE of a city. Invasion ships have a specialty of city attack, just as PDS's have a specialty of city defense>
It's the combination of ships in your "fleet" (attack stack) that make it most effective, just as in real life naval units (and most other civ games)