A Short Intro
WARNING: Much of the content of this guide is very explicit, so much to the point that I describe what boots are for. If you wish to ignore the explicit material, please glance only at the colored words, and imply the rest. Keep your hands and feet inside the guide at all times. Thank you, and enjoy the guide.
Start with your W and max it first because it gives you a nice DoT, extra damage boost, and greatly reduces the effects of enemy healing pots and support heals. Great for harrasing, for you can just run up and hit them once, potentially taking out a fourth of their health (depending how squishy they are), or just for farming minions.
Your Q comes second in priority, for now the enemy can't avoid you as well because you have a gap-closer. You can activate your W, Q to the enemy, appply your "poison", maybe slap them around a bit, and leave. You can also dash back towards your tower and/or team if you get thrown to the wrong side of the enemy lines, dealing damage on the way.
I put E in third position, because it is used mainly for dodgeing, and you don't need to up the damage on it untill end game. You don't need it beginning game, for there isn't much to dodge, and what little there is to dodge doesn't do much damage.
There are really a lot of good options here for Fizz: Ghost, Ignite, Clairvoyance, Heal, Surge, Teleport, and even Smite for jungling-- I go with Exhaust and Flash for a few reasons.
First off, Flash is good to get in there for a kill, or to get away while your E is on cooldown. Exhaust is good to slow down people, so that you can catch up to them, or get away from them. However, I tend to save my spells for a bit more selective circumstances.
Flash I usually save for double-jukes. For example, if I'm running thru the forest for some reason, and get ganked by someone I can't take, I'll use my E to get over a wall. If the enemy doesn't have flash, you're usually good. However, if they do have flash, they could follow you, in which case you jump the wall again with your flash.
Now, if the person ganking you has a movement ability of their own, say, Ezreal, they can do the same thing. Here's where you have the advantage. Certain walls can be jumped over with Playful, and then jumped back over with Trickster. If you're getting chased by an Ez, just run to one of these walls, double juke with your E alone, and triple juke with Flash. It's real EZ.
Exhaust is for occasions in which you are fighting 2 or more people. Exhuast one of their DPS, and take out the support (or other DPS) while the exhaustee is exhausted. They are now 70% less of a threat to you. Quickly dispatch the target, then focus the one who was just exhausted. If you're quick enough, they might even still have exhaust on them, and so they are slower and have 10% less Armour and Magig Resist.
Amplifying Tome: Why do I start with this? The answer is really quite simple: Damage. Both instant, and over time. With the runes and masteries I assign, you end up starting out with 30 ability power, and adding the amplifying tome leaves you with 50AP right off the bat. This is especially useful with the 9.4 Magic pen you start with, taking out a third of the enemy magic resistance, resulting in about a 27% increase in magic damage.
Health Potion: I play very aggresively as Fizz, and so he does end up taking damage from minion waves and the skills of enemies trying to get away.
Sheen: More ability power, helps to fix the mana problem on Fizz, and an extra 70 Damage applied on or after your Q. Perfect for early game kills. I always get this the first time I go back, so long as I can afford it.
Boots: This is what I get the first time I go back if I can't afford Sheen. Allows you to get back to your lane faster, chase the enemy champion back down their lane if they pushed, dodge skill-shots, escape ganks, and just help you get around the map faster (I.E., for ganks).
I upgrade the boots to Sorcerer's Shoes simply because of the extra magic penetration that is needed, and of course for the increased movement speed.
Fiendish Codex: I get this first in Morrelo's for the cooldown reduction. You should have your shark soon, or already have it, and being able to use it 10% more often is really nice, along with reducing the cooldown on your E by 2 seconds.
Morrelo's: I choose Morrelo's over Rod of Ages because it gives more immediate AP, mana regen, and a very nice amount of cooldown reduction. It's also more reliable than Rod of Ages, because there's no time limit to have to get it by (in case you're having a really bad game), or amount of time you have to have it (in case you're having a really fast game).
Rabadon's Deathcap: Because you can never have too much AP. This gives just about the most AP in the game, as well as boosting the AP of all your other items. It greatly increases your damage and sets you up nicely for your next item.
Lich Bane: This is your best item. Once you get to this point, you become pretty much unstoppable. Loads of ability power, decent magic resist, and a 7% increase to movement speed. If that's not enough, you are now dealing an extra 455 Magic Damage with your Q, or simply with auto-attacks. This MELTS people. But wait, there's more! You don't have to spend as much money on this itme, because you already paid for half of it with your starting item! Take that, Doran's Ring. (Note: I am highly against starting with Doran's ring on Fizz, you can't afford to waste that much gold, and your health potion should take care of any healing that needs done. I'd rather get a RoA than Doran's ring.)
Void Staff: Now we're just to the icing on the cake. This item gives a good chunk of ability power, but I get it mostly for the 40% Magic Penetration. With this, you pretty much seal the deal against tanks.
Will of the Ancients: Congratulations, you're a team player! If Void Staff was icing on the cake, then this item is freaking sprinkles. Besides giving loads of AP to yourself, your teammates also get 30 ability power, and 25% spell vamp. With this much spell vamp, your Q will pretty much full-heal you. Check and Mate.
I get Magic Pen Marks because 8.55 Magic Pen results in about 24% increased daamage at level 1, and that goes a long way.
Armour Seals are for early game harrassment, and make up about 20% of your total armour at lv 18. They protect mostly from minions (as well as your passive), and also from enemy champions' auto-attacks.
Ability Power Glyphs and Quints are to help give you an early game damage boost, so you can get that first blood and get fed. All together, these runes increase your AP by 23.76. That's like another Amplifying Tome at lv 1.