Laning Phase
Fizz has a very weak laning phase against the majority of typical middle lane champions pre level 6 as he will not have much killing potential, nor the ability to poke. It's important to stay in lane as long as possible, only going in to last hit and using potions liberally. A good technique to do on Fizz is to play very aggressively in the lane phase, expending all of your mana to push in your lane to the enemy turret. You will take a lot of harass, but this will let you go back to fountain and purchase more potions and items than your lane opponent. This is only good against enemy heroes that can't clear waves very well at tower early on however, such as Swain. You can go in to harass your lane enemy if they use their spells to last hit i.e. if an enemy Brand uses his W (his AoE circle skill) to clear creeps, you can activate your W and use your Q to go in on him, as he won't be able to trade with you without his full skill set. This makes it a much easier ordeal to try and kill him once you reach level 6. Once Fizz hits level 6, he has the power to zone most mid lane champions from the creep wave. This is because if he lands his ult, he will often deal enough damage with the rest of his skills + Ignite to pick up a kill, nor can most mid lane champions kill Fizz because of his projectile dodging ability (Veigar, LeBlanc, Brand). By pushing your wave into the enemy tower, you are given the freedom to roam around the map or force fights at Dragon. By ganking a lane, you can help your teammates get an advantage over their lane opponent and get back to your lane without missing many creeps, or you can force a Dragon fight which will mean that your lane opponent will lose the giant wave of creeps at his turret, or lose Dragon. If it becomes risky to start doing Dragon as your team groups around it, your team can all disassemble, which will still result in your enemy losing all of the creeps at his turret. If a fight ever breaks out, the general progression of your skills should be to use your ultimate on a squishy if you can; else, use your Q to get into melee range of a squishy champion. Then you should use your ultimate and Ignite, hopefully getting a kill or at least fighting until you are low enough / forced to use your Playful Trickster to run from a fight.

If against a champion who Fizz excels versus such as Karthus (squishy and Fizz can juke his Lay Wastes) or Katarina (squishy and melee), you can opt to play much more aggressively, hitting them with Seastone Trident a couple times when they go in to last hit, and then chasing them off to their tower with an Urchin Strike. However, be wary of junglers who aim for early ganks such as Lee Sin who can kill you with ease, especially if you have taken Heal.

Midgame
During midgame, you should continue to try and force fights or gank around the map, utilizing Fizz's strong midgame presence. You should try to end the game early, as Fizz tapers off late game when the enemy team groups up and enemy carries begin to build magic resistance items. A really good trick is to leave your lane, pretending to be going around ganking and sit in a brush in the river. If your enemy chases after you thinking that you have gone to gank a lane, they'll end up walking right in front of you leaving you in melee range of them, letting you pick up an easy kill. Always remember that wards are a great investment in that you can get vision of an enemy squishy wandering alone, which can lead to an easy kill and followup tower kill or Dragon kill.

Endgame
Hopefully the game has not gone long enough to the point where the enemy team winning is still a threat around the ~25 minute mark. The reason that Fizz becomes weaker lategame is because as soon as he leaves his E or is caught out of position, he will die instantly without Hourglass, and only survive slightly longer with Heal. You can circumvent this by building defensive items, but I believe that building defensively means that you don't have enough burst to be much of a threat to an AD carry. Rather than starting to initiate full 5v5 team fights, you should try to catch an opponent in the jungle, split push, or wait for them to dive your team / start Baron and then fight them when they are at a disadvantage.

TACTICS

In fights, you should stay back and try to protect your AD carry if you can't jump on top of an enemy squishy without overextending and having to use your Flash or Playful Trickster. Although Fizz's primary role is to assassinate enemy squishy champions, your team will win the fight if the enemy carries are playing safe and not dealing very much damage in fear of you. Otherwise, you should Q over to the enemy squishy champions, using your R and Ignite to hopefully kill them.

TIPS AND TRICKS

You can jump over walls with both your Q and E.
Fizz's ultimate is strange in that you not only have to decide the direction it goes in, but also the length. This means that it's entirely possible to shoot an ultimate that would have hit someone at maximum range, but end up aiming it short and hit nothing but terrain.
You can combo your E with your flash, and surprise your enemy. You can flash while coming down for trickster to put your damage elsewhere. Similarly to a shen taunt, before coming down you can Flash to gain more range, damage the enemy and reach him unexpectedly.
SYNERGIES AND COUNTERS

Synergies
Because Fizz is a melee champion with no ranged harass, he is completely useless in 'poke' situations, where both teams trade harass each other with damage without committing to a full blown team fight. Thus, good champions to complement Fizz are champions that are very strong at starting team fights.

JARVAN IV

AMUMU

MALPHITE
Counters
Because Fizz relies on surviving his laning phase and snowballing early in order to be of use in the game, champions that counter him in lane make it very difficult for him to be of use in a team setting.

DIANA

SWAIN

ORIANNA