First Rule: Fizz is over all Weak. Always keep that in mind.
Second Rule: your W has a lot of Damage, On hit and Over Time.
Third Rule: your E is not an engaging tool. it is for escapes or total annihilation.

Site Note: Your Damage spikes are at LvL 3-4 then at LvL 6-11 after LvL 11 you are pretty week till the Rabadons kicks in.

Pros First Cons Second
+Huge burst -Easy to kill once Combo is used
+Great Escape -No Escape if E is miss used
+Great engage -Relies on his R to win Team Fights
+Good-ish Early -Good Early IF u have some experience on him
+Hyper Late game carry -Really bad Mid game, Once u finished LichB and Zhonyas u Zhonyas u lack a lot of DMG till your +Once rabadons is out you can Rabadons is up.
Kill the carrys with one combo
+Ignite and escape

Laning is easy Against AP Caster Champions since your'e gonna start with your E and u should be able to dodge. but like i said Rule 1 Remember That Fizz is weak. don't pick fights before LvL 3. At Lvl 1 try to Dodge and let your E hit them while u land behind your minions. DON'T DOUBLE-PRESS E, the hit Box after you re cast your e is a joke so don't use it except u gotta go somewhere fast. The best think o do is wait for LvL 3 stay under the tower and try to get some few minions under the Tower, Once you hit LvL 2 go for your W so u can farm better under the tower if needed. Finally LvL 3 so what we are trying to do now is winning Trades. How do we win trades? We Activate our W while we are casting Q we try to get 1 AA of and usually leave. so your Trade combo should look like this Combo Number:1 "Q+W+AA and escape IF NEEDED with your E trying to avoid damage, Now the important Part If u play against a Caster like Brand,Zed,Anivia,Annie etc. and they have wasted most of there damage u try to land to use this Combo Number:2 "Q+W+AA+AA+E" but this time u engage with Q+W and start hitting them Second rule applies here "your W has a lot of Damage, On hit and Over Time" so this is where we deal the most damage in lane trading if u can get such a good trade try to keep the lane dominance they should be at LvL 4 by now and have used at least 1 Pot u try to Spam those Combos which one is yours to decide. You basically try to spam combo N:2 if possible or at least combo N:1 if u get them to around 40% HP they should be an easy target. Now how to kill them, try to get close Enough to use your Combo N:2 what is important is that once u engaged and start Auto Attacking they will try to flash out usually after the Second AA they will try to flash or once u casted your E now the nice thing is u can Flash while on your E so your early lane Execution combo is this :Q+W+AA+AA (After the second AA it should be an easy kill so we use our Ignite to secure it) +Ignite+E (Now get ready to flash if they flash) +AA there u got First Blood at LvL 3-5.

As a Fizz u can Start a Good Teamfight try to stay in the back and try to flank Always try to engage with your R, try to hit one of the Squishy carries ADC low HP Mages If this succeeds try to follow up with Q+W or a Flash+Q+W if needed. If they have an over all squishy team you can also go R+E+AA+W+Q+Zhonyas in if u are sure u can wreck them.

If u Have any further questions or need explanation about something just write it down.
Thank u for reading my guide.