Fizz is cool, and dominates the lane against almost anybody, he can burst people down so fast they will not even know what hit them. He is very mobile and it is easy to catch people or run away, with hopping over walls or dashing to minions.

As Fizz your objective early game is to farm as much as possible, with the help of Seastone Trident you would be farming more faster than most AP mid champion.
If your sure that you can win a trade go for it, harass as much as possible so that you will be able to push your lane and help other lanes, but also keep in mind to save some of your mana. If your having trouble with your opponent (example mordekeiser) make sure to ask for ganks more frequently. And since Fizz needs early kills in order to do well late game, you should always try to kill your opponent, if your getting a hard time getting kills, first option would be to just play passively and farm , second option is to ask help from your jungler, also remember being risky with Fizz can sometimes be good(study shows that being risky with Fizz can be rewarding, , you will never be able to succeed or get a kill if you are too scared.

Mid game usually starts when players start to roam and when the outer turrets have been destroyed, at this point, Never ever go alone, and always stay with your team mates. Also take farming as a priority and always ask for the blue buff. Also little skirmishes will be happening often so it's better to always have your Playful / Trickster up so that you can go in and out of team fights. Your objective here would be to immediately end the game, since as the game progresses your enemies would be able to catch up with your farm and kills, and may even win the game. But when you are the one whose losing, always tell your team mates to Never Give Up.

At this point you would be able to easily kill squishy targets and push. . And if you have been following my instructions, you would have already farmed up well and you have almost completed your items, So the main thing to do here is to avoid dying, and avoid being ACED!, and ward dragon and baron more frequently, and always stay with your team mates no matter what.

Now, in team fights fizz is quite good if you use playful/trickster to get to the backline of the fight and burst down their adc, and then zhonyas so you cant be targetted and your playful/trickster will be off cooldown you can jump back out.