Since Fizz can move through units, find opportunities in lane to walk through minions and apply your 'Seastone Trident' ability - the damage will rapidly add up as the opponent's Health drops.
Fizz's ultimate ability, 'Chum the Waters', can be aimed at an enemy or towards the area where you think they'll be going.
Fizz's spells scale off Ability Power - consider getting items like Rod of Ages or Rylai's Crystal Scepter against teams that have high burst threat - and items like Rabadon's Deathcap if you think you can survive without the Health.
Fizz becomes extremely deadly when your Health reaches critical levels - each of Fizz's autoattacks apply a damage over time effect that rapidly grows in damage as your Health becomes lower - stay healthy when fighting Fizz, if possible.
Fizz is best in extended skirmishes - don't chase Fizz if you can help it!