Fizz is a very unique champion and by far very entertaining to play. His ability kit makes him great from early game to late game. He is a very versatile champion and can be built many ways and played as many different roles, but for this guide, it will be for AP mid.

Pros
Very bursty when played AP mid
Great CC between Chum the Waters and Playful / Trickster
A fun and rewarding champion
With Seastone Trident you can burst squishies as well as do good damage to tanks
Good versatility and many effective builds.

Cons
Abilities are close-ranged, so can cause bad positioning
Fizz is melee, so early game CSing in mid can be hard against a lane bully
Playful / Trickster has a relatively long cooldown, so can generally only be used offensively or defensively

The masteries I use are pretty standard for an AP mid. Run up the offense tree with 21 points toward the AP masteries for Executioner then put 9 into defense towards Veteran's Scars . This will give you some good base HP for early game to sustain better and the offense tree will provide good damage.

If possible, I will start with a Doran's Ring for a 50 AP start at level 1. A lot of times, you can pick up first blood with a Doran's Ring start from the damage output after you get all your abilities, but if you think you are going to get bullied alot (remember you are melee so they can poke you easily when you farm) start with Boots and 3 Health Potion. After you go back, work towards a Sheen. If you happen to snowball alot before your first time back, pick up a Crystalline Flask for some lane sustain. After your Sheen, finish the Sorcerer's Shoes and start working towards a Rabadon's Deathcap. If you are having troubles CSing because of a lane bully or if you are snowballing hard, grab a Kage's Lucky Pick. The extra gold generated will help you late game while also providing a good amount of AP. After your Rabadon's Deathcap, you will be a great asset for your team. The AoE ultimate with a knockup while dealing substantial damage will be very noticeable in team fights. After Rabadon's Deathcap, finish up your Sheen into a Lich Bane. After your Lich Bane, you will most likely become a focus for the enemy team. I usually like to build the Rod of Ages because it is a great item and gives you some HP, mana, and AP. After that, depending on their composition, I like to build some defenses. I usually grab an Abyssal Mask to get MR and magic pen since you are usually close enough to your opponents for the aura to come into play (which lowers MR). Then after that, for my final item, I like to grab either a Guardian Angel or a Zhonya's Hourglass. If you think you could use some more AP, go with the Zhonya's Hourglass, but if you are getting focused and insta-killed in team fights, consider the Guardian Angel. Feel free to adjust your late game items for different team comps.

Many Fizz players will tell you to max Seastone Trident first because of the bleed damage, but I usually go with my Playful / Trickster first if im playing AP mid. Maxing your Seastone Trident is good, but since it does a percent of missing health, I believe it is good enough to hold off until being maxed later. I max my Playful / Trickster first because this gives you great AoE damage and slow. At level 5 the slow from Playful / Trickster is very good. As always, level your Chum the Waters whenever you can. While you are in the laning phase, you want to focus on poking your opponent down without taking too much damage in return. After you have poked your opponent down, you go in for the assassination. During the early game, use your Urchin Strike to charge in on your opponent and activate your Seastone Trident. Your Urchin Strike applies on-hit effects so your opponent will be tagged with the Seastone Trident passive. After this, you can either go with more damage from your Playful / Trickster or back off. A lot of times when you charge an opponent, they will automatically start backing away towards their tower, so you can use your Playful / Trickster and land the slow, then just back off without them being able to keep up. This is a great way of harassing. Then once you get your Chum the Waters, you become very bursty. Remember that it has travel time, so lead your shots if you have to. Once you land your Chum the Waters, you'll want to Urchin Strike in and use your Playful / Trickster while they are slowed from Chum the Waters and knocked up, making it very easy to land your slow once again on them. Then while they are slowed dramatically, you'll want to activate your Seastone Trident and auto attack them until your bleed can finish if they are still alive. In team fights, try to land Chum the Waters when they are grouped to knock up and slow as many as possible. Don't be afraid to become the focus in the team fight with your Playful / Trickster off of cooldown. It is a great escape tool and you can use it to get out while taking some aggro that your teammates could have died from. After they get low you can Urchin Strike back in and assassinate.

The summoner spells you choose can vary. With your Seastone Trident bleed, Ignite isn't necessarily as important for Fizz than other champions. If you want to make sure you secure kills, feel free to grab Ignite, otherwise, going with something such as Exhaust, Ghost, or Teleport is good as well. In my opinion, Flash is a must have. Fizz is an exceptional turret diver and Flash just makes it even more viable. With Flash you can dive an opponent under turret with Urchin Strike and still use your Playful / Trickster for the great damage and slow while still being able to Flash out instead of not being able to use your Playful / Trickster offensively.