First off I really don't play summoners rift anymore and no longer play my ranked account, So my advice might be "meh". But when i did, Fizz was always my main pick. I am going to assuming you don't care about picks bans and team composition, so I will just ignore that. The truth is fizz can almost always be poked out of lane and for the first part of the game you are often trading health and mana for cs this is normal (try and make smart trades). Now that doesn't mean you will lose early and be down cs you can still win lane and win early.
The real advice for trading in lane actually starts here
So no brainer here if you can get some good all in ganks you should win lane.
The key to trading in lane and csing without just getting crushed against a smart opponent is watching their skills and trying to trade when key skills are down, or also trying to trade by juking a key skill with playful trickster, if you can juke the skill and land on or better yet slightly behind them perfect, this can also be used to cs. example Tibbers not up but Annie still holding onto that stun just for you, bait it out playful trickster right behind her urchin strike through her (or to a minion away assuming she stunned you with the other spell since even if you juke the q stun she gets to keep the stun... for some bull**** reason).
Capitalize on any mistakes they might make know what they can do to kill you. example That LB just use w to clear a wave and trying to harass you with q, juke it and hit her she's used 2/5's of her all in as long as she's pushed out and you're not being ganked you should win the trade.
Also don't be afraid to use abilities to cs (do it as sparingly as possible since your mana pool is tiny)
Now for Yasuo, he is actually a pretty difficult opponent for fizz, when I was still playing summoners rift I only encountered a few. From my experience he is gonna shove and its gonna suck. It's much easier to fight him outside of a minion wave and you need to pop his shield before you all in (not true if you're really fed). Honestly I find catching him and killing him isnt a big deal but laning against him was very difficult and if you have trouble trading while laning just try and keep your cs up. More current fizz players may have some outplays for him but this is all i have for you.
this is all the advice i have some of it might be outdated but most of this actually works despite the champ you are playing as, just different abilities used in a similar fashion.