This is my play style and set up for Fizz, some may like others won't, hope you enjoy!

For my runes I mainly go for ability power, some may want mana regen and in that case a Greater Glyph of Replenishment would work just fine. As you know Ability Power = more damage and burst.

I go for Magic Resist in this build because there is always gonna be that pesky AP magic damage champion that can two shot anything if played well. This is great for countering just that. The boots give a nice boost early game and allow you to poke more with your Seastone Trident and Urchin Strike at level 1 and 2 without worrying too much about a lot of burst. Late game is when you need to worry, especially in team fights. Athene's Unholy Grail gives CDR (Cool down Reduction) which is great for Fizz and gives Mana Regen on top of the resist. I highly recommend rushing it if you're getting nuked early game, picking up Chalice of Harmony first and then Fiendish Codex. Lichbane is strong with Fizz because with his Seastone Trident and Urchin Strike you can nuke someone under their tower and then Playful/Trickster and get away.

With the Armor Build this is to counter champions like Tryndamere and Master Yi, which we all know can be very hard to kill late game. When you get your Zonya's your not only getting a decent amount of armor but an insane on use, use this when Tryndamere is low with ultimate up and wait for it to fall off, use it to leap into a tower for a kill and to asorb that one tower hit that would otherwise kill you. Rod of Ages and Rylai's help you from getting bursted from 100%-10% with one decent combo, if you are still getting hit hard I would recommend getting Randuin's Omen or Warmog's over Rod of Ages.

I pick this Skill Sequence because it is unlikely you will be using Playful/Trickster for anything but getting away. Seastone Trident is his main source of damage, it makes his auto-attack worthwhile and when he uses the active you can hit pretty hard with a good Urchin Strike and land early kills. His Urchin Strike is good, but not as good as Seastone Trident at level one, you can poke and let the dot tick a little, then go in again and with his passive this is easily done even through minions. Try not to use ultimate on just one person unless it is completely needed, or unless you are about to get kill early game and you know you can land it.

Pros

High Mobility
Good Burst damage
Good survivability
Very good in team fights with ulti

Cons

Decent mobility
Medium damage
Bad survivability
Not useful in teamfights

For efficient farming on minions make sure you get your Playful/Trickster at level 3 and use it to last hit all minions in the area.

Well that's it for my Fizz guide, I hope you learned something useful. Good luck with games to come as Fizz!