Code:

Pros

+ Great assassin + Annoying + Makes the enemy ragequit when ahead + Simple kit + Fish + Can towerdive safely

Fizz is an extremely fun champion once you get used to him, very mobile and very strong early to mid game. Very satisfying to one-shot enemies with. He's a yordle, so he's annoying already, a global taunt, which can work to your advantage.

Cons

- falls of super-late game - Weak when behind - Squishy - Melee - Can't farm safely because all his abilities are short ranged - He's a yordle

Our beloved fish suffers from his short-range late game. He also suffers from smart enemy players who actually build against us, banshee is one of the best items right now, it gives lots of ap and Mr and that annoying shield that blocks one ability, this forces fizz to either waste his ult just to delete the shield, or get into melee range aka danger zone, to do any damage.

As you approach the first wave, don't level up any abilities yet. Most times you don't even want to auto any minion at all. See if the melee minions all get low at the same time, so you can finish them off with one e. If that's the case, level up your e and finish the minions with an e. If you hit the enemy laner at the same time, that's great. Make sure you don't double tap e, otherwise, you can lose the minions and you won't slow the enemy even if you land it. Depending on the enemy laner, you can even weave in an auto or 2 to proc electrocute.
If the minions aren't lined up to die from an e, you can take w at lvl 1 and just farm with it. Finish your minions with an auto+w and the w will reset and give back mana and cooldown.
At level 2, level up either w or e, whatever you didn't take before. See if you can make a quick trade on the enemy laner with e auto w to proc electrocute. Even at lvl 2 this combo can take almost half hp of your laner.
At level 3 is when I try to see if I can all in the opponent. Most times I can, but even if I can't, I try anyway, maybe I burn a flash or make them back. Make sure you don't farm with e, just with w. Only use your e and q for the enemy.
If you manage to kill your opponent or make him back to base, shove the wave and back and buy and reset. Rinse and repeat.

Mid game is a great time for Fizz. I call mid game the moment you finished up your first 2 items and your boots. Most probably there's a tower or two down too. This is the moment any squishy near you is one shark away from the gray screen. Fizz is a great splitpusher too, he melts towers with his w, he can run away if necessary and can dive if need be. Don't be afraid to dive if you are ahead, you can reset and avoid tower shots with your e. Simply q auto w to proc electrocute and e out.

The late game is kinda meh for Fizz. It's not that he's weak, it's that there are lots of champions who just scale better, have more range and by now probably a defensive item that makes them hard to one shot.
You still have the power to oneshot, but squishy champions and adcs can pretty much kill you before you even get to them. That veigar that you used to oneshot now bought a banshee or zhonya and he can survive your initial burst and you are vulnerable to the rest of their team. That kaisa that you used to one tap 10 min ago, now she's always invisible and too mobile for you to even click on her,that free kill caitlyn now kill you in a auto and a headshot before you even get into range to eqw her and even if you shark her, she has guardian angel. Also ad champs who bought edge of night, now are hard to kill.
TLDR Fizz suffers from lack of range late game and from the fact that late game there are too many 5 vs 5 fights. Fizz is not that good in 5 vs 5, especially late, he's better in 2 vs 2 or 3 vs 3. He also sucks against champions who bought banshee, edge of night, zhonya and other ******** items that prevent you from killing them in a rotation.

Basic combo in lane is q-auto-w-e. After you e, it's up to you if you need to e on top of enemy for damage or e away to safety. It will come with experience. Of course if you enemy is low and you want to all in, drop on him with your e for the slow(don't double tap e, or will not slow him) and then run him down with autos and ignite.

Another combo in lane, is to q a minion to get closer to your laner, then e on top of him for the slow, auto + w and either finish him off or back away.

Best way to use your e is to use it when the enemy throws an ability on you and at the same time e on top of them for damage.

After you get ult, don't be afraid to throw it max range, they will either burn flash or die, or at least force them out of lane.

Low enemies won't be able to hide under tower, since you are one of the few melee champions who can dive safely. Simply q-auto-w the enemy laner and e right after to avoid the tower shot. If you think the e + ignite will kill them, drop on them and just walk out of tower range. You will get a tower shot or two, but the enemy probably didn't have the opportunity to deal any damage back. Flash away if you think tower towers shots will kill you. With experience, you will learn how to dive properly.

One combo you need to know is the e + flash. When you e, right before you drop from your pole, you flash on top of the enemy. You will mostly use this to follow after they try to flash your e to avoid getting killed.

Another sneaky combo is the q+ult, it's one of the most undodgeable skillshots. You q a minion and as you travel with your q, you ult the enemy. It's extremely uncommon for this combo to be dodged, but it does happen if you are really far away, if you are close, it's almost impossible, unless the fizz is playing against me, since I'm expecting it whenever the fizz has ult up. This guarantees ult will hit. After that all in for the kill.