When laning as a fizz, you want to be farming well or else you will fall behind. When you see a gap where your laning opponent is weak or vulnerable, use your combo w-q then jump away using e. After level 6 you would start off by using your ult (r) then doing the same wombo combo w-q then finishing with e if opponent is low or to get away. It is essential to build health for early tankiness, like a Rod Of Ages then building into a lich bane to put more damage onto your Q.

While playing fizz you are a big key to your team, you are the bursty assasin who should be taking out the enemy carrys. If you need to engage with you ultimate to secure a kill for your team. Other than that you shouldn't be engaging unless YOU HAVE TO. You should be sitting out on the sides/back waiting for a time to go in and then get the hell out. I like to say "In and out like Batman" :)