This guide is under heavy construction since i just recently started playing Fizz.
This guide is my take on Fizz and is not something you should follow like a bible.
Feel free to comment if you have anything to add.

Fizz is a melee/mage fighter with good Crowd control and mobility. He is pretty weak the first few levels so dont get to cocky in the begining.
His skills are extremly mana heavy so use them wisely.

Full damage runes, if you feel that you need more mana due to the fact that his skills are very mana heavy switch seals to mana regen. This is something i have to experiment on aswell.

Masteries in offense tree are self-explanatory.
Masteries in defense tree is for the lack of early game health due to the fact that i do not use Dorans ring.

Start with the standard boots and potions. get the sorc boots first and then move on to RoA, get the Catalyst part for extra hp and mana.

When RoA is done start building your Nashor's Tooth.
This is THE item for fizz, it has everything he needs, especially cooldown reduction.
since we wont be using flash we want trickster to be on as low cooldown as possible.
With this build your trickster will be on a 5,68 second cooldown and Urchin strike on a 4,26 second cooldown.

I have not yet decided which item is best after Nashor's but deathcap or lichbane would be preferable.

End with Malady for even more AP, AS and magic resist reduction

Playful/Trickster is maxed first because you want it on as low cooldown as possible for the good mobility it gives, think of it like a small flash and Vladimirs Sanguine pool combined.
Next priority is Urchin strike because we want that on as low cooldown as possible aswell.
Ofcourse take your ultimate whenever you can (6,11,16).